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\ 5^ 
.^1 



HENRY FRO'*DE, M.A. 
K0HL18HTEH TO THE C HIVKH8I T Y OF OXFOHIJ 
LONDON, EDINBURGH 
NKW YORK 



emm in ehglmwj. 



TO LADY LOW 



My dear JFriend, 

But for your resolute energy I think I 
should never have communicated to the educa- 
tional world the method which I have long used 
for reviving the faculties of children suffering 
from mathematical rickets and logical paralysis. 
These diseases are common ; they are induced 
by the practice of teaching mathematical pro- 
cesses on a hypothesis about the nature of 
Mathematics directly opposed to that which 
underlies the original invention and formulation 
of these processes. 

I have, as you know, taken no part in recent 
attempts to improve methods of teaching, be- 
cause, as James Hinton said, I can see no use 
in trying to invent good methods for teaching 
truth on a basis of falsehood ; the only remedy 
in which I have any faith is to tell the pupil 
one or two simple laws of the relation of the 
human mind to Scientific Truth, and then to 
see that he forms the habit of working in 
accordance with those laws. 

For nearly forty years, authorities of various 
kinds have been assuring me that it would be 
impossible to do this except in individual cases ; 



4 DEDICATION 

that the public will not tolerate being shown 
the glowing heart of the mathematical dis- 
coverer, will not dare to let itself know to what 
virgin inspiration it is paying homage when it 
confers medals and honours on an original 
mathematician. 

Your Ladyship has intervened in the situa- 
tion, refusing to see objections and difficulties, 
or to take ' no ' for an answer. You insist that 
the public shall at least have the option of 
deciding for itself whether its children shall 
know what lies at the heart of genuine Mathe- 
matical Science. I must therefore ask you to 
accept your share of the responsibility of offer- 
ing to parents and teachers what I fear is after 
all only a very feeble presentation of Arithmetic 
treated as a branch of the Art of Thinking, 
founded on the general Science of the Laws of 
Thought. 

Yours affectionately, 

M. E. Boole. 





• 


CONTENTS 








PAGE 






7 




Introductory Address to Young 








29 


I. 


How Men Learned in old Times 


31 


II 


On Counting by Tens 


37 


III. 


Why we do not always do Sums 






the way that comes Natural . 


45 


IV. 


Arithmetical Shorthand . 


52 


V. 


Keeping Accounts .... 


56 


VI. 


DlS-MEMBERING AND Be-COLLECTING . 


58 


vn. 


Weights and Measures . 


65 


VIII. 


Multiplying by Minus 


69 


IX. 


Adding Minus 


75 


X. 


Dividing and Sharing 


78 


XI. 


In what consists Economy 


81 


XII. 


Economy op Mind-Force . 


87 


XIII. 


Exercise in Kelevance; Introducing 






the Idea op Probability . 


94 


XIV. 


Exercise on Zero .... 


97 


XV. 


*-°° 


98 


XVI. 


Equivalent Fractions 


101 


XVTL 


Greatest Common Measure 


106 


xvni. 


Standard Weights and Measures . 


117 



6 CONTENTS 

PAGE 

XIX. What can be settled by Human Law 122 

XX. Paper Money 127 

XXI. The Dog's Path . . . .132 

XXII. The Ball's Path .... 136 

XXIII, Exercise to prepare for General 

Formulae 139 

APPENDIX: On Proportion. ... 142 



PREFACE 



The motto chosen for the title-page of this 
little work may seem unsuitable as an Intro- 
duction to a course of lessons on Arithmetic ; 
a subject which to many persons seem9 so emi- 
nently un-heavenly and dull. But then, the main 
reason why it seems so is that their teachers 
failed to put them in possession of that c Re- 
covered Past,' the bearing of which on the 
Present forms the great clue to that knowledge 
of subtle forces which gives its possessor the 
key of the Future. 

The present little volume is not intended 
to interfere with ordinary methods of teaching 
Arithmetic, or to supersede the books already 
in use. It can be used under any School system 
and in conjunction with any text-book. Not 
more than one chapter is intended for use in 
any one term ; the earlier chapters are suited 
to little children, the later ones for children of 
fourteen or fifteen. The lesson for the term 
may be repeated each week with varying illus- 
trations. 

Teachers of such subjects as Electricity com- 
plain of the difficulty of getting pupils to apply 
what they know of Mathematics (at whatever 



8 



PREFACE 



level) to the analysis and manipulation of real 
forces. It is not that the pupil does not know 
enough (of Arithmetic, or Algebra, or the Calcu- 
lus, as the case may be), but he too often does not 
see, and cannot be got to see, how to apply what 
he knows. Some faculty has been paralysed 
during his school-life ; he lacks something of 
what should constitute a living mathematical 
intelligence. In truth he usually lacks several 
things. 

In the first place, though he knows a good 
deal about antithesis of operations (e.g. he 
knows that subtraction is the opposite of addi- 
tion, division of multiplication, movement in 
the direction minus — x of movement in the 
direction x, and so on), he has not the habit of 
observing in what respects antithetic operations 
neutralize each other, and in what respects 
they are cumulative ; and surely no habit is 
more needed than this as preparation for 
making calculations in electricity or mechanics. 

In the next place, he too often knows, 
about the idea of relevance, only enough to be 
foggy about it. The reason for this is that his 
study of the idea of relevance itself began where 
it ought to have ended ; his attention was never 
called to it till the stage was reached when it 
would have been right that he should direct his 
action in regard to it sub-consciously from long 



PREFACE 9 

habit, leaving conscious attention free for dealing 
with the actual elements of some question which 
is difficult enough to need thinking about. He 
was not made to grasp the fundamental idea 
that a statement may he relevant to one question 
and irrelevant to another, till some knotty prob- 
lem occurred involving consideration of which 
statements are relevant to the special question 
in hand. So he had to try to grasp at once the 
idea of relevance and the question, what is 
relevant to what, in a special problem. Such 
thrusting on the young brain of two difficulties 
of different kinds at once, is contrary to all 
accepted canons of Psychology. Examples are 
here suggested (Lesson XIII) in which there can 
be no doubt as to what is relevant to the ques- 
tion at issue ; the child's attention is therefore 
free to focus itself on the idea that there can 
be facts concerning a thing which in no way 
concern the particular question which is just 
now being asked about the thing. 

Then again, whatever skill he may have 
acquired in the manipulation of those notations 
and formulae which he has been taught to use, 
he knows hardly anything about the manner 
in which such things come into being. Now 
an applier of Mathematics to real forces should 
be able, when occasion requires, to modify his 
notation, or invent a new formula, for himself. 



10 



PREFACE 



He cannot begin to learn how to do this, straight 
away, while his mind is struggling with problems 
of electricity or mechanics ; he should have had, 
from the first, the habit of seeing through for- 
mulae and notations ; of watching them coming 
into being, of helping to construct them. 

The special difficulties found in interesting 
pupils in Arithmetic are in no way inherent in 
the nature of the subject ; they belong rather 
to the domain of medical psychology ; then- 
cause can be fitly described only in the language 
of that science. However logical may be the 
course of a teacher's demonstration, given his 
premises, those premises are usually introduced 
to the pupil under conditions extraordinarily 
unfavourable to concentration of the attention- 
power. One can only describe them by saying 
that if analogous conditions were set up in the 
stomach-nerves at the beginning of dinner, the 
result would be lessened appetite and digestion- 
power; if in the muscles at the outset of a 
cricket-match, the running would in consequence 
be slower, the batting, bowling, and fielding less 
accurate, and the heart-action more irregular ; if 
in the eye- and finger-nerves before a drawing- 
lesson, the result would be dazzled sight and 
trembling fingers. 

It may be objected that if this temporary 
partial paralysis is set up, in the Arithmetic 



PKEFACE 



11 



class, only on the occasion of the first presenta- 
tion of some new idea or principle, i.e. perhaps 
once in a term or two, it is not frequent enough 
to induce disease or permanent weakness of the 
attention-power. But if any condition of nerve 
disturbance or lowered vitality accompanies the 
first presentation of an idea or object, there is 
a strong probability that each subsequent con- 
tact with that object or idea will produce, by 
association, a recurrence of similar disturbed 
action. Thus is set up a habit of scattered 
attention which retards the development of the 
faculty of concentration, and makes any real 
grasp of principles difficult, especially in regard 
to Mathematics. One main object of the present 
volume is to present the premises involved in 
various arithmetical operations in a manner 
which concentrates instead of scattering the 
power of attention, and thus leaves the pupils 
fit to attend to the teacher's reasoning. 

My own experience, however, is that a child 
who has been helped to grasp the premises of 
a mathematical argument, in any hygienic man- 
ner, seldom needs much assistance in making 
out for himself the logical consequences. 

The Preface to a recently issued text-book of 
Arithmetic ' asserts that ' the young are always 
eager for a rule or formula which will save 
1 Arithmetic, Kirkman and Field. 



12 



PREFACE 



them the trouble of thinking for themselves.' 
The young who are reeling under a blow on the 
head from a bludgeon may not unnaturally be 
eager to be taken home to bed in the easiest 
vehicle at hand ; and there is no telling what 
habits of lethargy might not be generated by 
a systematic course of such treatment ; but the 
young who have been left in possession of their 
normal faculties prefer using those faculties in 
active personal exercise. 

Another teacher says ' : — 

'In stocks and shares the pupil is usually 
dealing with things the nature of which he 
does not understand. A little political economy 
must be drilled into him. Bring the meaning 
of the abstractions of the subject home to 
him. . . . The great difficulty is in problems. 
Nothing but stern practice can avail here. Dis- 
courage all formulae. Anybody who knows any- 
thing about boys knows that if, by any means 
under heaven, they can avoid using their brains 
and attain the same result by a mechanical 
application of a formula they will do so. They 
cling to a formula as to a straw, in the ocean 
of new ideas in which they are drowning.' 
Exactly so ; children's minds are too often left 
to clutch at straws ' in an ocean of new ideas.' 

1 Workman, ■ Oxford Junior Local/ in Educational 
Jteview. 






• 



PREFACE 



13 



If, two or three terms before a pupil has to do 
with fluctuating values, he had any clear idea 
why a Bank of England note has a fixed value, 
it would be easier than it usually is found to 
make him grasp the meaning of stocks and 
shares. 

The mechanism by means of which man 
learns is intended to play among the concep- 
tions Unity, Negation, and Fraction ; Number 
is less cognate to it. The kind of fractions dealt 
with in Arithmetic books is, it is true, too 
recondite to be thrust on a brain at an early 
stage. But if mathematical faculty is to de- 
velop normally in the youth, the child should 
have early practice in dealing with the idea of 
a unit broken into bits, which Jit together to 
make up a whole. 

Nature provides savages with opportunities 
for dealing with this idea, to an extent of which 
it is difficult for us to form any adequate con- 
ception. Something has or has not been caught 
for dinner ; and when it has been caught it 
has to be shared and is entirely eaten. Next 
day the child again sees an animal of the same 
kind whole. The child's mind plays con- 
stantly between the idea of rabbit (or whatever 
animal may be used for food) whole, rabbit 
shared, no rabbit; and again, whole rabbit. 
This has a very different effect from what is 



14 



PREFACE 



induced by the sight of a loaf or joint appear- 
ing as a matter of course ; a loaf from which 
every one is helped, the rest being taken away 
when no longer interesting. This is one of the 
reasons why an existence uniformly, uninter- 
ruptedly prosperous is not always the most 
favourable to intellectual development. 

In all ranks of life, it is true, infants provide 
themselves with opportunities for exercising 
the faculty of reconstructing a unit from its 
fractions, by tearing or breaking things and 
fitting the hits together again. Our Kinder- 
garten systems to some extent provide for 
systematic exercise of the same kind by means 
of various toys. Even there, however, though 
much practice is given in constructing, too little 
is given in re-constructing a unit out of its 
fractions. But in schools the whole provision 
for this practice, which is the essential fibre of 
all mental growth, seems left to chance, and 
too much crowded out. 

In Arithmetic, where it is specially important, 
hardly any opportunity is afforded for practice 
in swinging the mind between the conceptions 
Unity, Negation, and Fraction. The mass 
material of Arithmetic itself (i.e. the art of 
dealing with numbers) is packed into the mind 
artificially ; little or no possibility is provided 
for it to build itself up by natural accretion 







round its organic supporting fibre, 
to induce a condition of what I 
mental rickets. 

It is desirable to fix children's minds on the 
act of Negation as a positive act of mind, to 
a far larger extent than is usually done. Oppor- 
tunities for this might be taken in connexion 
with sums in multiplication and long division ; 
the zeros should be written in, as statements of 
fact, for some time after the pupil has grasped 
the idea that their omission does not actually 
affect the ultimate answer. 

Use might also be made of such exercises as 
those of Chapter XIII. 

Again, the teaching of Arithmetic is much 
facilitated, if, besides actual exercises on o and I 
and very simple fractions, logical exercises 
parallel to the operations of Arithmetic and 
constructed on the same models, are carried on 
in that borderland region where mental opera- 
tions parallel to those involved in Arithmetic 
deal with questions of Art, or such simple 
portions of History, Ethics, or social relations, 
as come fairly within the scope of a child's 
intelligence. Each operation of Arithmetic may 
find its analogue in one of these borderland 
exercises in Logic. 

It is of primary importance that children 
should clearly distinguish between what they 




16 



PREFACE 



do or recite because they are told to do so, and 
what they themselves grasp or see. They must 
do certain things because they are told, and 
commit to memory facts which they will need 
to know ; but they should always be taught to 
distinguish between : — 

' I repeat this because I was told,' and ' I see 
that this is so.' 

Some of the worst mental habits are induced 
by the practice of teachers making a statement 
as if ex cathedra, and then proceeding to bring 
forward proofs of its truth ; this is Euclid's 
method, but Euclid apparently wrote for grown- 
up men, perfectly well able to take care of their 
own minds ; he wrote probably for the best 
intellects of his time. His method is unsuited 
to children. If a teacher has anything to say 
to children as a statement, he should say it, 
not exactly as dogma which they are bound to 
believe, but as working hypothesis which they 
are to assume as a basis for the present. Any- 
thing which he intends to prove should never 
be stated ; children should be led up to find it 
out for themselves by successive questions. No 
pains should be spared to keep the two sets of 
statements apart. 

To return to Mi-. Workman's complaint of 
the abuse of formulae, surely the remedy lies 
not in discouraging the use of them, but in 




PREFACE 



17 



encouraging children to use them intelligently 
and to make them for themselves. They should 
begin early the practice of entering certain 
kinds of results in a book. This book may be 
divided into two parts: anything specially 
needful to remember, if found out by the 
children, should be entered in one part ; if 
told them by the teacher, in the other. 

The sentimental people who assert that every- 
thing in Arithmetic can be ' proved ' to children 
have, usually, no idea of what rigid proof means ; 
it is not necessary that the child should see 
the evidence for every hypothesis on which he 
works ; what is necessary to mental health is 
a clear understanding of what constitutes evi- 
dence, and the power to distinguish between 
what is, and what is not, proved. 

The children should be encouraged to use 
freely, for reference, any tables or formulae 
constructed by themselves. This should be 
permitted as an indulgence, a labour-saving 
luxury, the reward for accuracy in recording 
the results of previous investigation. In private 
teaching and in very small classes, no child 
should be allowed to enter any formula into 
the form-book till he is individually able to 
work it out unaided. In ordinary class-teach- 
ing this may be impossible. In that case the 
process should be gone over at intervals, till 



18 



PREFACE 



the class as a whole can, amongst them, evolve 
every step. The formula should then be entered 
by each child in his own form-book ; and the 
impression should be conveyed that it is the 
communal work of the class, the expression of 
its conjoint logical investigation. 

The formula-book should begin with a blank 
form of multiplication-table, which is to be 
filled in by the child as practice in addition. 
The operation of multiplying should be intro- 
duced to the child's mind in some such way as 
this : — ' Add together four and four and four 
and four and four.' When the addition sum 
has been done correctly, say : — ' You will have 
to add five fours together very often. You 
may enter the result in your table-book ; and 
next time I give you five fours to add together, 
I will allow you to look for the answer in your 
book instead of adding.' 

All the modern higher Mathematics is based 
on a Calculus of Operations, on Laws of 
Thought. All Mathematics, from the first, 
was so in reality ; but the evolvers of the 
modern higher Calculus have known that it 
is so. Therefore elementary teachers who, 
at the present day, persist in thinking about 
Algebra and Arithmetic as dealing with 
Laws of Number, and about Geometry as 
dealing with Laws of Surface and Solid Con- 



PREFACE 



19 






tent, are doing the best that in them lies to 
put their pupils on the wrong track for 
reaching in future any true understanding of 
the higher Algebras. Algebra deals not with 
Laws of Number, but with such laws of the 
human thinking machinery as have been dis- 
covered in the course of investigations on 
number. Plane Geometry deals with such 
Laws of Thought as have been discovered by 
men intent on finding out how to measure 
surface ; and solid Geometry with such 
additional Laws of Thought as were discovered 
when men began to extend Geometry into 
three dimensions. The branch of Mathematics 
called Quaternions deals with such Laws of 
Thought as reveal themselves during the pro- 
cess of investigating the relations between n 
and n + 1 dimensions. The sooner pupils are 
made to see all Laws of Arithmetic as Laws of 
Thought, not of things, the simpler and more 
satisfactory will their future course be. 

This is especially the case with regard to the 
sign — ; it should never be interpreted in any 
such way as to convey the impression that it 
indicates negative quantity, subtraction, or 
diminution of numbers or of things ; but 
always as indicating something about the point 
of view from which the things or numbers are 
considered. Neglect of this caution lands one 

B 2 




20 



PREFACE 



in endless metaphysical contradictions and 
absurdities. Such for instance as that a rect- 
angular surface becomes = o if we happen to 
place the origin of co-ordinates in the middle of 
it ; for it then divides naturally into the. four 
rectangles X x Y, X x ( - Y), ( - X) x (- Y), 
(— X) x Y. The amount of confusion and 
hindrance to progress thus induced is sickening 
to think of. 

Teachers seldom observe how early many 
children are affected by such anomalies as 
these, or how profoundly they are affected by 
them. If the same boys who are thus pain- 
fully impressed became teachers in their turn, 
they would no doubt remember their own past 
experience, and a better mode of presenting 
the primary conceptions of Arithmetic and 
Geometry might soon be evolved. But, un- 
fortunately, these logical thinkers are usually 
repelled by the impossibility of extricating 
themselves from the network of insincerity, or 
what seems to them such ; give up the study 
of Mathematics, and betake themselves to some 
other pursuit. Very much of the waste and 
confusion which gather round the sign — might 
be obviated by making children go through such 
exercises as those given in Chapters VIII and IX. 

Much needless difficulty is caused by the 
attitude of Mathematical teachers towards the 



PREFACE 



21 



symbol oo , which they often interpret as some- 
how connected with endless length or exhaust- 
less quantity. The effort to conceive of either 
endless length or exhaustless quantity is a 
metaphysical gymnastic very unhealthy for 
a young mind. It sets up also a set of ideas 
and connotations quite irrelevant to any use of 
the symbol ce in the higher Mathematics. 
c© has reference not to length or quantity, 
whether great or small ; but to release from 
certain restrictions to which the values specified 
as finite have been subject. (See Lesson XV.) 

The whole terminology connected with the 
operations called in Arithmetic books * Multipli- 
cation and 'Division' is excessively misleading. 
The words ' multiply' and ' divide ' might have 
been invented on purpose to create confusion in 
passing from integral to fractional Arithmetic. 

The continued use of the word ' dividend ' 
especially, in Arithmetic, is a symptom of the 
carelessness of teachers in the matter of avoiding 
causes of confusion. In old days, when most 
people who had money hoarded it in the house or 
deposited it in a bank, till they could use it in 
some business of their own, the word 'dividend' 
had no connotation for a child's ear, till it was 
explained to him as ' an amount to be divided.' 
But in these days of limited liability companies, 
the word ' dividend ' is incorrectly but very 



22 



PREFACE 



commonly used, at home in the hearing of 
children, to indicate a sum of money which is 
the result, or quotient, of the division and 
distribution of the year's profits among the 
shareholders : ' I will do so-and-so when my 
dividends come in,' &c. Even if the teacher 
Ls careful to explain that the word in the text- 
book does not mean what it means in common 
parlance, which not all teachers remember to 
do, that explanation in itself causes distraction ; 
some word should be chosen which has no mis- 
leading associations. Of course children ' out- 
grow ' any misapprehension which may be thus 
caused, and learn that ' dividend ' has two 
meanings. But one wonders what teachers 
suppose is the effect on the mental habits, on 
the brain tissue, of struggling, for even a few 
days, to understand division before this dis- 
covery has been made ! The present writer 
has not authority to devise a nomenclature ; 
and therefore in this introduction employs that 
in common use, though under protest. It 
should be avoided in speaking to the class. 

The teacher should, in dealing with any 
branch of the operation called ' multiplication,' 
have in his mind this idea : — ' Multiplication ' 
means doing to tho operand (multiplicand) what, 
if done to unity (1), would produce the operator 
(multiplier). 






PREFACE 



23 



In dealing with ' division ' he should keep in 
his mind the idea : — ' division ' means doing to 
the operand (dividend) what, if done to the 
operator (divisor), would produce unity (1). 

The operation of division should be introduced 
at first, not as the inverse of multiplication but 
as a succession of subtractions, thus : — 

If five boys are to share twenty apples among 
them, how many can each boy have ? Let us 
give each boy one apple first ; that takes off? 
Five apples. And leaves ? Fifteen. Now we 
give each boy another apple : that takes off? 
Five more. And leaves? Ten. Another to 
each boy leaves? Five. Another to each 
leaves ? None. So now we have finished. 
How many times did we go the round and 
give each boy an apple? Then each boy has? 
Four apples. 

A little practice in such successive sub 
tractions should be gained, before the idea is 
introduced of doing long division straight off. 
Division should not be treated as the mere 
inverse of multiplication, but as a way of cutting 
short a laborious series of subtractions. The 
child should be led to see that the multiplica- 
tion table can be used to cut short not only 
series of similar additions, but also series of 
similar subtractions. A great deal of the diffi- 
culty afterwards found in dealing with such 



24 



PREFACE 



operations as G.C.M. is traceable to neglect 
of this simple precaution. 

The manner in which that most logical of all 
text-books, Euclid, has been misused to induce 
illogical habits of mind, repeats itself hi most 
departments of elementary Mathematics. A 
good instance of such misuse can be shown in 
connexion with the subject of equivalent frac- 
tions {Lesson XVI). Let us suppose that the 
teacher wishes to prove to the class that three- 
quarters of one is equal to one-quarter of three. 
He takes as his unit of thought the concept 
apple. He too often begins by stating a thing 
which is not true in itself: viz. that three- 
quarters of an apple is the same thing as a 
quarter of three apples. For the purpose with 
which the teacher's mind is occupied at the 
moment (the equivalence of fractions), the state- 
ment is time ; it is as true as that, from the 
shop-keeper's point of view, twelve pence are 
a shilling. But the child, who has not yet 
been introduced to the conception of equivalence 
of fractions, is at a point of view quite different. 
To the child's imagination 'great lots of grub ' 
have a fascination quite independent of the 
size of his own share. To suggest (for instance) 
that a bun for one child is the same as a hundred 
buns for a hundred children would be to insult 
the pupil's understanding and his feelings — 



PREFACE 



25 



feelings which affect him all the more keenly 
that as yet he does not know how to express 
them. If he could believe you, what would be 
the meaning of school treats and social picnics? 
In proportion^ as you ultimately induce him to 
feel that one bun for a child is the same as 
a hundred buns for a hundred children, he will 
be, in futurfe, the worse man, citizen, political 
economist ; and the less fit to apply mathe- 
matics to problems in real forces. But we are 
speaking now of the effect of such statements 
on his intellectual processes, at the age when 
his sensations are as yet un warped. The state- 
ment that a quarter of three apples is 'the 
same as ' or • equal to ' three-quarters of an 
apple has brought prominently into the field of 
his imagination two pictures : — an apple, and 
a group of three apples. The natural way of 
making an apple equal to three apples is to 
bring forward two more apples. It is the way 
things used to be made equal when he was 
learning addition ; why not now ? The teacher 
has therefore started him looking out for the 
two more apples ; and as what is said next does 
not seem to be tending in that direction, his 
attention is distracted ; he gives only half his 
mind to what is said. Thus he fails to get hold 
of one or more links in the chain of argument, 
£oroe children soon recover themselves, and 



26 



PREFACE 



attend to the rest of what is said. And, as 
the teacher seems to be satisfied with the 
reasoning and to expect them to be so, they 
imagine they are so. This helps to form a habit 
of intellectual dishonesty, of confusing sham 
proof with real proof. The children who are 
either less sharp or more logical and thorough, 
look longer for the non-appearing two apples, 
and miss more links of the chain of proof. 
These children feel that the whole demonstration 
has passed in that region where grown-ups 
conduct a self-satisfied mental life in which 
children cannot share. Thus are formed habits 
of 'hopeless non-comprehension/ perhaps even 
of ' self-protecting and contemptuous non-atten- 
tion.' The chapter on equivalent fractions 
(Lesson XVI) suggests ways of attacking the 
problem not open to this objection. 

It may occur to some that too much use is 
made of examples relating to food. But we 
cannot make mathematicians by insisting upon 
a non-existing superiority to physical facts. 
Apple or bun forms the natural unit for a child ; 
the sharing of a cake or fruit is the natural 
fraction as well as the true introduction to the 
higher ethical life. As a matter of fact the 
Arithmetic of grown people is largely occupied 
over questions of food supply and of personal or 
family interest. To place children's ideas about 



PREFACE 



27 



such subjects on an honest basis and give them 
a social direction would surely tend more to- 
wards ethicalizing the community than the 
setting up of topics of fictitious interest, and 
then teaching children to be satisfied with 
reasoning' which is either imperfect in itself or 
but partially understood by them. To let the 
children be thoroughly logical about their own 
sphere of action among things which they do care 
about,and deal practically with duties which they 
must perform, is far better training than to en- 
courage them to express opinions about subjects 
the data of which lie beyond their personal ken. 
The pauses of sitting with slack muscles and 
taking slow quiet breaths, to make mind-pictures, 
is not intended to supersede the usual permitted 
' intervals ' between lessons ; they are au integral 
part of the Logic lesson itself. They, as well as 
the rest of the scheme here suggested, form, it is 
believed, the first attempt to adapt to elementary 
education here the magnificent method of study 
described in Gratry 's Logique. A certain element 
of normal mental life, essential to sound intel- 
lectual development, and which can only be 
received during suspension of the active per- 
ceptions of intellect, is being driven out by 
modern educational grind. This tends to 
produce a sort of mental and moral rickets, 
analogous to the physical disease induced by 



28 



PREFACE 



a diet deficient in the bone-forming element. 
Teachers who have studied any sort of medical 
psychology bitterly lament that the require- 
ments of existing systems force them to inflict 
this grievous wrong upon the rising generation. 
In the present volume, an attempt is made 
to re-introduce at least a small share of the 
skeleton-forming elements of mental life, in a 
manner which does not necessitate disturbing 
existing arrangements. The author has been 
kindly assisted in this matter by Mrs. Archer 
and her pupil Miss Cross, of Coombe Hill School. 

English children have now to be taught to 
relax, to breathe, to see. It is probable that 
the cultivated art of relaxation will ultimately 
prove more satisfactory than the mere following 
of natural impulse to relax. 

But it is certain that the teachers who have 
weeded out the wild growths must see about 
planting the cultivated ones. Such exercises as 
those here. recommended tend to promote sane 
and healthy mental action, and to make the 
discipline of school more organic and harmonious. 
They have a great fascination for children, as 
my young friends at Coombe Hill School know. 
I am sure they will be glad that I am attempt- 
ing to give to other boys and girls a share in 
the fun- that we had over our little sums. 

Mary Everest Boole. 






INTRODUCTORY ADDRESS TO YOUNG 
TEACHERS 

Some of you may perhaps have seen what are called 
Japanese Flowers. You buy a penny packet of things 
which look like rather clumsy wafers, each of which 
has a fine line round its outer edge. You put one of 
the wafers into warm water, and (as the printing on tho 
envelope says) ' Watch the result.' The line round 
the edge melts and slips off. Then perhaps, from the 
wafer come out first one little coloured flower, then 
another, and another. Or there may come out leaves, 
or sea-weed. The change seems wonderful, but the 
explanation of it is simple enough: some one in Japan, 
by great care and skill, made each leaf and flower ; then 
they were linked together by stalks made of thread, and 
then the whole set was rolled up into a tight form 
convenient for carrying about, and fastened round with 
a little strip of paper. And we have to put them into 
warm water which acts as a solvent (that is to say, 
.1 melter), before we can see how pretty they are. 

Arithmetic seems to some people dry and un-bcuutiful ; 
but that is because they have not soaked it in that 
solvent which is called sympathy. If we had sympathy 
with the struggles and labours of others, Arithmetic 
would be easier to understand and pleasanter to learn 
than many children find it. 

It must, however, bo distinctly understood that any 
of the following lessons are to be omitted or modified, 
if, in the judgment of the teacher, it is unsuited to the 
condition of the class. 

This remark applies especially to Lesson I. If the 
discipline of the claws is is such condition that the 



30 INTRODUCTORY ADDRESS 

teacher would not find it easy to restore order after 
a burst of laughter, he may suggest to the children, in 
some quieter way than that here given, that men held 
up all their fingers, as a way of saying ' Many,' before 
they had learned to count how many. But my own 
experience and that of several of my friends is that the 
dramatic introduction of the savage element, where this 
can be safely permitted, has a wonderful effect in 
conquering the apathy of wliich so many teachers 
complain, and making Arithmetic real and living. This 
remark applies to children of all ages, and at all social 
levels. 

In this book you will find no long sums. The 
following caution may be found of use in dealing with 
those which you Bet from the ordinary text-bookB. 
Children should be carefully taught to suspend attention 
at proper pausing-plaees ; as, for instance, at the ends 
of lines, in multiplication, long division, practice, or 
equations. Just as, in reading aloud, one must take 
breath at suitable stopping-places, or one will be driven 
to stop for breath involuntarily at the wrong place, so 
it is with the attention-power iu Mathematical work : 
if it is not trained to take a moment's rest at the right 
place, it rests itself (or, as we say, ' the mind wanders ') 
in the middle of a line of figures ; a wrong figure is 
written down during the lapse, and sets the whole sum 
wrong. This acts very unfavourably on those children 
who have specially short-breathed attention-power ; who 
are often, otherwise, the cleverest, and might become, if 
properly trained, the most intelligent mathematicians. 
On the other hand, a training in the proper care of the 
attention -power is good for all children alike. 



HOW MEN LEARNED IN OLD TIMES 



You have heard, ray dears, that, a long time 
ago, there were no people in this country except 
savages. Perhaps some of you have read that 
delightful book, TJie Story of Ab ; or you may 
have seen the picture of a savage with his wife 
and baby, at the beginning of Clodd's book, 
Primitive Man. 

How do you think children learned lessons in 
those days ? Do you think Ab's wife ever told 
little Mok to wash his face and hands', and come 
to lessons ? Do you think that the hairy woman 
in Clodd's book ever called out to her boy : 
1 The school bell is ringing ; make haste, or you 
will be late ' ? Oh ! no ; nothing of that sort 

1 [In old Miracle Plays the first woman is represented 
an calling her boys in to wash their hands and brush 
their hair, because to-morrow will be Sunday and they 
will he examined on the Catechism. The future villain 
of the piece initiates his evil career by some such breach 
of good manners as taking off his cap with his left 
hand ! These and countless other literary phenomena 
of the same kind go to show how far astray the unin- 
structed imagination goes in its presentation of historic 
data ; and how necessary it is to take frequent oppor- 
tunities of pulling it into line with the reality of things.] 



32 



LOGIC OF ARITHMETIC 



ever happened at that time. Yet the children 
did not grow up quite ignorant. We are going 
to have a little talk to-day about how people 
learned in those days, and what sort of teachers 
they had to make them learn. 

Why do the monkeys in the Zoological Gar- 
dens like climbing on their ropes ? Why is it 
found good for the health of children to climb 
the gymnasium ladder and cling to the bar by 
their hands ? Why do most of you enjoy doing 
so as soon as you have got a little practice at 
it ? Why do dogs not climb ladders or ropes, 
and why is it not found good for their health to 
teach them to do so ? You and the monkeys 
had ancestors who climbed trees in order to be 
out of the way of wild beasts of the dog kind. 
These ancestors of yours forced themselves to 
learn to climb, by great labour and effort, driven 
by great fear and desperate need. But they 
went on, generation after generation, trying and 
trying, till it came quite easy and pleasant, till 
they could climb for pleasure and fun. And 
that has worked into the very muscles of your 
flesh and the blood of your veins, the power and 
the wish to climb. But the dogs seem to have 
had no ancestors who learned to climb ; when 
their ancestors were afraid of enemies, they 
protected themselves in some different way. 
Primitive Man climbed trees quite easily, and 



HOW MEN DID LEARN 



33 



80 'lid Ab, and Ab's wife, and Mrs. Primitive 
Man ; they carried their babies up trees till the 
babies were old enough to learn how to go up ; 
and then the mothers made the children learn. 
And all that has helped to make climbing 
natural and good for children now. 

But another thing was happening at the same 
time. Have you noticed what sort of words 
a child learns first ? Names of things, or names 
of numbers ? For instance, if a child of about 
a year and a half old saw six birds on a wall, 
would he say ' Six,' or would he say ' Dickie- 
birds ' ? You know that he would say 
' Dickie-birds ' much earlier in his life than he 
could count ' six.' Well, savages of very long 
ago, such as Primitive Man, could not talk as 
we do ; indeed, one name for them is ' Speech- 
less Man.' But they must have had words 
or grunts or growls or signs of some sort for 
some few things ; even a hen can tell her 
chickens that she has found food, or that she 
sees a hawk. We may be sure that Primitive 
Man could tell his wife that there was a wolf 
near when he could not say ' Six wolves.' Per- 
haps he said ' Woo-oo-oo-oo' by way of a sort of 
imitation of a wolf's howl. 

And presently he began to find out that 
it would be convenient if he could somehow 
tell his wife whether there was one wolf or 



34 



LOGIC OF ARITHMETIC 



many. He had no words, as yet, to tell her 
with. 

Now what does an omnibus conductor do, 
when you have heard, or guessed, that he said 
1 Fares, please 1 ' ; but, because of the rattling 
noise, he cannot make you understand whether 
your particular fare is two pence or three ? He 
holds up fingers. Can we not fancy Primitive 
Man doing that ? If he said his word or grunt 
for wolf, and held up one finger, that might 
mean : ' Just stand behind me with baby, my 
dear, while I kill the wolf with a stone.' But 
if he spread out all his fingers, that might 
mean : ' Let us get up the tree quick ; for there 
are more wolves coming along than you and 
I can manage.' 

We see now that the savages of long ago had 
teachers ; though they had neither desks nor 
books nor school bells. Outside of themselves 
there were wild beasts which taught them to 
be sharp and try to escape danger. Inside of 
them they had feelings which taught them that 
it is not always enough to climb up a tree by 
oneself, that they must help each other and 
take care of the baby. And, between the two 
sets of teachers, they somehow learned to say 
1 One ' and ' Many ' on the fingers of their hands. 
Neither set of teachers could have taught them 
alone ; neither the wolves outside them nor the 







feelings inside. But between the two sets, the 
savages learned their lesson so well that, to 
this very day, we have never forgotten it. 
Children count on fingers still, unless there is 
some reason for using counters instead. 

Poor Mrs. Primitive Man, and all the rest of 
our wild ancestors ! I often think of them when 
I am puzzled and worried myself. They little 
guessed what delightful gymnastics they were 
preparing for you children, or what a convenient 
way of doing sums they were helping to pre- 
pare for us all, by their care of their babies and 
by their brave struggles against laziness and 
cowardice. Perhaps, thousands of years hence, 
people will be enjoying some wonderful science 
or delightful art, of which we now have no 
notion at all ; and will be looking back with 
thankfulness to the struggles and troubles of 
us now, which are preparing for their enjoy- 
ment. 

Now before you go to the next lesson and 
forget this one, I want you to make a picture in 
your minds that you will remember in after life. 

Shut your eyes; lean back comfortably in 

your seats. Let your hands lie quite slack on 

your laps. Take a few long easy breaths. Now 

then. 

(The Teacher should read the following with a paute 
between oach sentence.) 

C 2 






30 



LOGIC OF ARITHMETIC 



Think of the funny hairy man in Clodd's 
book. See him start suddenly. He calls out : 
' Woo-oo-oo-oo.' The little ones may call out 
1 Woo-oo-oo-oo ' if they like. Now be quiet again. 
He calls out ' Woo-oo-oo/ and puts up his hands 
with all his fingers spread out. His wife comes 
out of her dreamy mood. Jumps up and carries 
her baby to the foot of the tree. Clasps the 
baby between her long hind feet. Climbs the 
tree like a monkey by her hands only. The 
man climbs after her. When they have reached 
a high branch, she takes baby in her arms : she 
and the man sit side by side, and wait till the 
wolves have gone by. 

(Two or three minutes ahould now be spent in silence, 
sitting at ease, before the class breaks up. Attention 
should be paid to relaxation of that muscular tension 
which accompanies all active attention to external facta. 

Modern children are fast forgetting the secret of 
slow, deep natural breathing, as well as that relaxed 
attitude and meditative picture-soakage which ib one of 
Nature's most powerful educators. The loss of these 
things is probably one groat factor in the failure of 
modern educational schemes. 

As the sudden rush from active attention to one topic 
to active attention to another is in itself bad for the 
young brain, the last few minutes of each class seems 
a good opportunity for practising concentration. 

The above method for impressing mind-pictures on 
children is an adaptation, suited to ordinary class ti; idl- 
ing, of a more elaborate method carried out at Coombc 
Hill School.) 




What is this, 1 ? And this, 2 ? And this 
3 ? (and so on), and this, 9 ? 

Now I want to write ten : how shall I do it? 

Put 1 and 0. 

But what has ten done to be different from 
the rest ? 

Why should it have two signs instead of one 
like its neighbours ? and why does it take signs 
belonging to its neighbours, instead of having 
one of its own ? 

Did ten ask to have two signs ? Did it wish 
to have two ? No ; then why do we give it 
two? 

I once asked a young friend of mine why he 
did something in his sum ; and he answered : 
• My reason is that I was told to do it at school ; 
but I know I ought to have another reason, and 
I know I haven't.' I thought that was a sensi- 
ble answer. It applies to most things in 
arithmetic : your reason for writing ten with 
two signs is that your teachers told you to do 






38 



LOGIC OF AEITHMETIC 



so ; their reason for telling you to do so is that 
it has been found a convenient and suitable 
way for people to write ten. But there is 
a reason why it is suitable and convenient, and 
you ought to know that reason. All your suras 
will seem to you more interesting, and, I think, 
easier, when you know it. 

We have seen under what sort of pressure 
people must have found out how to tell each 
other that there were one or many wild beasts. 
But when once they had learned that much, 
they could apply it further. By-and-by they 
began to keep flocks of sheep or goats, and they 
made stone walls to fence in the sheep by night, 
so that wolves could not get at them. Perhaps 
they might wish to count their flocks, to see if 
all the animals were safe in for the night ; or 
one man might wish to sell his sheep to another. 
Well, at first the owner might count by putting 
up a finger for each sheep, as it passed in 
through the gap in the fence. That did very 
well up to ten. But what next ? How could 
he show anybody that there were sixteen sheep 
or twenty-nine sheep ? He might put up all his 
lingers for ten, and then put them down ; and 
then put up six fingers for the other six sheep, 
that would mean sixteen. He might put up 
all his fingers twice for two tens, and then nine 
fingers ; that would mean twenty-nine. 




ON COUNTING BY TENS 



39 



But if he had many tens of sheep, he might 
lose count of the number of times he put his 
hands up and down. What was to be done 
about that ? 

The man who had very many sheep to count 
must contrive somehow to make a mark for 
each time he put his hands up. 

There was no paper in those days, no pens or 
pencils, not even balls strung on wires such as 
we see in some infant schools. I believe people 
had not even names for more than a very few 
numbers. There are still people in the Malay 
Islands who have no names for any numbers 
except one, two, and three. People had to 
think about keeping flocks in order to be sure 
of having food and clothes for their families, 
before they began thinking about what was the 
best way of doing sums ! What they wanted 
was to make sure somehow that all the sheep 
they sent out in the morning were safe inside 
the fence at night. There was a quicker way 
to manage that than inventing figures, and 
finding out how to make pencils. The man 
could call up his little daughter, who had no 
school to go to, and say to her : ' Would you 
like to help Daddy count the sheep? Just 
stand nice and quiet beside me, and I'll tell you 
presently what to do.' Then each time he 
finished putting up all his fingers, he would 



40 



LOGIC OF AEITHMETIC 



make her put up one finger, till all the sheep 
had passed out. Then he would look well at 
his own fingers and the little girl's, and make 
a picture on his mind of how they looked. 
Then he would perhaps say : ' Now, dear, 
when the sheep come home we must get our 
hands looking just like this ; and we must not 
be satisfied till they do.' In the e.vening, if his 
hands and hers looked as they did in the morn- 
ing, he could take her in to supper, and have 
a game of play with her, and then go to bed 
quite happy, because all his sheep were safe in 
for the night. But if either he or she had too 
few fingers up, and no more sheep seemed 
coming in, he knew that he must send her in 
to her mother, and call his big son, and go out 
on the moor to look for the rest of the sheep. 

Now I think we had better spell a number 
in the way that the shepherd and his child did ; 
then you will know better what sort of way it 
was. Tom, come here please ; stand at the 
right hand of the class ; you shall play at being 
the shepherd. Jane, you stand at the left side 
of the class ; you shall play at being the 
daughter. I cannot let the rest of you rush 
about pretending to be sheep, in school; you 
can do that in the play-ground if you like ; but, 
in school, I will say ■ sheep,' and you children 
must try to make a mind-picture of sheep going 



ON COUNTING BY TENS 



41 



out through a gateway, one by one, as fast as 
I count. Tom, every time I say ' sheep,' put 
up a thumb or finger. Jane, wait till I tell 
you what to do. Now, Tom : — 

££ One sheep ; two sheep ; three sheep ; four 
sheep ; five Bheep ; six sheep ; seven sheep ; 
eight sheep ; nine sheep ; ten sheep. 

Now you have no more fingers to put up. 
Hold your hands up high, and show the class 
that you have no more fingers left to count 
with. Put your hands down. Jane, put up 
one finger; that means that Tom has put up 
all his fingers once. Hold it high up for 
a minute, for the class to see that you have one 
finger up. Now, you need not keep your hand 
so high any longer, it would make your 
shoulder ache ; but keep that finger out 
steadily all the time till I tell you to leave oft*. 
Now Tom : — One sheep ; (repeat ££•). 

Hands down, Tom. Jane, put up one more 
finger to show that Tom has gone over all his 
fingers once more. Tom, one sheep; (repeat $*•). 

Hands down, Tom. Jane, one finger more. 
Tom, one sheep ; two sheep ; three sheep ; 
four sheep ; five sheep ; six sheep ; seven 
sheep. 

Now, both of you, hold your hands high, 
so that the class can see them. How many 
fingers has Jane got up ? Three. And each of 



42 



LOGIC OF ARITHMETIC 



the fingers means? That Tom has counted 
ten on his fingers. So Tom has counted ten 
three times ; how many does that make ? Ten 
three times ? Thirty. That is right. But 
Tom has done something else besides putting 
up ten fingers three times ; he has also put up 
seven fingers besides. How many sheep has 
he counted then? Thirty-seven. We will 
pretend that this is early morning and the 
sheep have just gone out to feed. Look well 
■t the hands. When the sheep come home at 
night, the hands must look just so, or else the 
shepherd cannot go happily to bed ; he must 
go and see where the missing sheep are. 

If you play that game out of school, you can 
play at counting other numbers ; forty, fifty, 
seventy, ninety sheep. But that would take 
too long now. 

For many, many generations people had only 
fingers to count with ; and, all that time, one 
finger of some other person may have stood for 
ten fingers of the person counting. So, at last, 
when people found out how to write, it came 
natural to them to make one in the left-hand 
place of the paper stand for ten in the right-hand 
place, just as it comes natural to monkeys and 
boys to climb things, because their forefathers 
had done it for so long that they did it without 
much thinking about it. It seemed easier to 



ON COUNTING BY TENS 

do it that way than to think about any other 
way ; the inclination to do it so had got worked 
into the very marrow of their brains. And so, 
when they came to want to reckon more than ten 
tens, they put the figure for ten tens in a fresh 
place ; and that is what we call the hundreds' 
place of a row of figures. And so they went 
on and on, making fresh places for the different 
powers of ten (as some people call it) because 
their great-grandfathers had ten fingers to 
count with. 

Now we come back to the questions we 
started with just now. Why does ten have 
two signs belonging to other numbers instead 
of one sign of its own ? It does not ; it has 
one sign. Its sign is a ' 1,' written in the tens' 
place, and that means : once the hands are put 
down, after counting ten, in order to begin 
counting over again. When we write twelve, 
which means ten- two, we write it so ' 12 ' ; 
that is, one ten or one double-hand, and two in 
the place of units as we call them. But if 
there are no things over, no units, we say so ; 
we put O in the units' place. That helps to 
remind the writer, and to show to other people, 
that the 1 stands for one ten, not one thing. 

If ever you are teaching a baby brother or 
sister to count, there are three things I should 
like you to be careful about. Never teach 



44 



LOGIC OF ARITHMETIC 



a baby to say ' one, two, three/ like a parrot ; 
always teach him to count things. If you have 
no counters, use some sort of blocks, or bricks, 
or collect little pebbles, or buttons. (Only 
then you must be careful not to let him swallow 
them.) Or he might count your fingers and 
toes, or the railings. But it is best to have 
things he can shift about, such as buttons ; and 
let him push each, as he counts it, from the 
uncounted heap to the counted heap. Never 
make a little child say ' eleven, twelve ' ; 
always tell him to say ' ten-one ; ten-two ; 
ten- three j ten-four ' ; grown-up people say 
1 eleven, twelve/ and now you are old enough 
to go to school, you must do the same ; but 
a little child should always say ' ten-one ; ten- 
two ' ; like ' twenty-one, twenty-two.' And 
as soon as he has counted ten things, make 
him put them in a little box or in a heap by 
themselves ; that will make him understand 
what he is about much better than most little 
children do. And then you can tell him the 
stories about Mr. and Mrs. Primitive Man, 
and the baby, and the wolves ; and about the 
shepherd and his little daughter* 



Ill 



WHY WE DO. NOT ALWAYS DO SUMS 
THE WAY THAT COMES NATURAL 

It is often best to do things the way that 
'comes natural,' as long as there is no reason 
for altering it. Very many things have to be 
altered as we become civilized and have to live 
in towns. It comes natural to children to count 
on their fingers, but they, have to be made 
to use counters or blocks instead, because 
counters can be arranged in convenient pat- 
terns : — 

Black-board. 



• • • • 

• • • • 



Whereas fingers are much less convenient. 

It came natural to bees to make round cells, 
because their forefathers had made little holes 
by rubbing their todies round ; but when the 
bees took to living a great many of them in one 




hole, they had to be stopped doing what was 
natural, and made to learn to fit their cells into 
each other, because it saved room and was more 
convenient. Poor little bees ! They didn't at 
first like learning the new way of making cells, 
I dare say ; but they had to. It comes natural 
to you to do some things that have to be stopped 
because they are inconvenient. It comes natural 
to a baby to grab with its hands at everything 
it wants ; but it has to be taught to eat pro- 
perly with a spoon. There are such quantities 
of things that we must alter, that it is generally 
best to go on doing things in the old way until 
a reason comes for altering them. It would 
take a good deal of trouble now to learn to 
count ordinary things otherwise than in tens. 
There is no particular reason why we should 
do so, so we all go on counting by tens ; and 
you were made to learn to do the same. Half 
the use of school is that children should learn 
which things they had better do the way that 
their forefathers did them, and which things 
we must now learn to do in some other 
way. 

Once upon a time a dear little girl about a 
year old was sitting on her mothers lap, and 
a bright teapot full of hot water was on the 
table close by. It comes natural to babies to 
try to get hold of anything bright that they 




SIMPLE ADDITION 

see. So this baby put out her hand towards 
the teapot. Mother said 'No/ and pushed 
the little hand away. But baby thought she 
would do what was natural ; she screamed and 
put out her hand again. Mother began to 
think that if baby was so anxious to get at 
the bright things, she might make a grab at the 
candle or the lamp some evening, or crawl to 
the tire, and have a bad accident. So she 
thought she would let baby have her way at 
once, when it could not do any real harm ; she 
pretended to give in, and let baby touch the 
teapot. It was very hot. It did not actually 
injure the child. But somehow, after that, baby 
understood why it isn't always nice and com- 
fortable to do just what comes natural ! 

You know that we must do sums in certain 
ways because we are told to do them so ; but 
also we ought, when we can, to have another 
reason. It is a good thing to play sometimes, 
when it does not matter, at doing a sum the 
* wrong' way, the upside-down way to what 
we have been taught ; then we find out why 
we were taught a way that does not seem 
natural. We are going to do that to-day. 

Suppose I have in my purse a shilling and 
two pence ; and some one pays me two shillings 
and three pence, how shall I write the account 
of what it ought to come to ? 




{ Three pence and two pence are , . . ' 

Stop a minute. Which does one care most 

about, the pence or the shillings ? The 

shillings. Well, why don't we reckon the 

shillings first ? ' It is the wrong way.' Why 

is it the wrong way ? Would it make the sum 

come wrong ? 

(Usually some children in a class think it would 
make the sum come wrong.) 

Well, we are only playing to-day, it does not 
matter if the sum does come wrong. Let us 
try. 

One shilling and two shillings make three 
shillings. Two pence and three pence make 
five pence. 

s. d. 

3 5 

That is the answer we get by beginning at the 
shillings end. Now let us try beginning at the 
pence end, as we are told is the right way at 
lessons. Two pence and three pence make five 
pence ; one shilling and two shillings make 
three shillings. So we have five pence and 
three shillings. Does that come to the 



111.-'- llllW 

same as 



SIMPLE ADDITION 49 

three shillings and five pence? Yes. And 
' three and. five pence ' is more natural to say 
than five pence and three shillings. Then why 
are we taught to do the sums in the way 
that is not natural, since both ways come to 
the same answer ? 

We must find out if we can. We must know 
the real reasons for things when we can. Let 
us try another sum : — 

Black-board. 
8. d. 

2 6 

3 8 



Three shillings and two shillings make five 
shillings. Eight pence and six pence make 
fourteen pence. How do we write fourteen 
pence? Two pence and one shilling. Where 
must we put the one shilling ? Along with the 
five shillings. So now our sum looks like 
this : — 

8. d. (a) 

2 6 

3 8 

~J 2~ 
6 

»OOL» D 




1 2 

Now let us do the sura the way we are shown 
at lessons. Eight and six are fourteen. Put 
down two pence and carry one shilling; one 
shilling and three make four; four and two 
make six. We have our sum looking so : — . 
s. d. (c) 

2 6 

3 8 



6 



2 



££ Now do you see why you were told to do 
addition sums in a way that is not the natural 
way ? Fancy if a shopman gave us our bills 
all crowded up with scratchings out and extra 
figures at every turn, like (a) or (b) I It is 
necessary that you should learn, while you are 
young, to do some things in the way that you 
would not think natural. Your teachers pick out 
for you which way you must do them in class ; 
they have experience and know ; but in holidays 
you may always try doing easy sums the upside- 
down way to the way you are told in 




i school. 



SIMPLE ADDITION 



61 



so as to gain experience for yourselves, and find 
out the reasons of the school rules. 

It is not safe to experiment, while you are 
young, with real things — with fire or food or 
your own bodies, because you might do harm 
that could not be undone. But sums can be 
wiped off or scratched out, when you get them in 
a muddle ; so they are capital things to experi- 
ment with, and great fun too; and you will 
understand your real work at school much 
better if you experiment in play. Always do 
your school-work, in term-time, exactly as your 
teacher tells you ; that is the way to grow 
smart and handy and useful. In holidays, do 
easy sums the upside-down way to the school 
way. That is the way to understand your 
work, and grow clever enough to find out things 
for yourself. 

At %< the teacher may interpolate other examples of 
beginning sinus at tho wrong end, taking an easy 
example of subtraction, multiplication, and short 
division. 

Look well at the black-board. Now Bhut 
your eyes and settle at ease. 

Can you 6ee in your minds the sums (a) and 

ybj f ^tuid any others tbat may have been dune in round- 
about ways). If not, open your eyes and look. 
Shut your eyes again. What do you see? 



D 2 




IV 



ARITHMETICAL SHORTHAND 



Black-board. 

12 Buttons are a dozen. 
12 Pence are a shilling. 

Are those statements true ? 

Are they both true ? 

Are they always true ? 

Are they true in the same sense ? 

Let us see. 

Suppose you go into a ^hop and say : ' I want 
a dozen of those buttons, please,' and a friend 
says : ' And I will take twelve of the same 
buttons, please/ do your two purchases look 
alike? Would the two lots weigh the same? 
Would the owner of one of the lots be any the 
worse or the better off, if the parcels were 
changed by accident ? If the twelve buttons 
were sewn on to your coat, would any one know 
that they were not the dozen ? If we wanted 
to play a game with twelve counters and had no 
proper counters, we might use the twelve but- 
tons for counters ; would the dozen do instead ? 
Yes, just as well. The dozen is twelve, and 
twelve is a dozen ; and for every purpose for 




AKITHMETICAL SHORTHAND 53 

which one could be used, the other would do 
just as well. 

You say twelve pence are a shilling. Do 
they look like a shilling? Are they the same 
colour, size, weight ? If I wanted things to 
use instead of counters, I might use twelve 
pennies : would a shilling do instead ? No. 
Sometimes in cooking, if we have not small 
weights, we use a coin as a weight ; we might 
be told in a cookery book to take the weight of 
a sixpence or shilling of carbonate of soda or 
of some spice. How would it be if we used 
twelve pence instead ? 

Twelve pence are not a shilling, not in any 
way like a shilling. Why do you say they are 
a shilling ? They are of the same value. Value 
for what ? Not for counters, nor for weighing 
things. 

Twelve pence are of the same value as a shil- 
ling when we want to buy things. Yes, now we 
have it right ; twelve pence are not a shilling, 
and cannot be used instead of a shilling for any 
real use of the things themselves. But the 
chief purpose of coins is to exchange for other 
things ; and, for that purpose, twelve pence are 
of the same value as a shilling. 

Then if that is what we mean, why don't we 
say so 1 Why do we say 'Twelve pence are 
a shilling,' when we don't mean it ? 



54 LOGIC OF ARITHMETIC 

Ah 3 why ? Now we have come to something 
which it is very important you children should 
understand and remember. 

Think how often, in your arithmetic, you 
have to make the change from pence to a shil- 
ling or from a shilling to pence. How would it 
be if the teacher had to stop eveiy time and 
say, ' It is arranged by law that, for purposes 
of buying and selling, twelve pence shall be 
considered as of the same value as one shilling ' ? 
The teacher would be tired of saying it ; you 
would be tired of hearing it ; it would waste 
time, and disturb you from attending to your 
lessons ; so it has been arranged that people 
may always say, ' Twelve pence are a shilling,' 
for shortness, whenever the business on hand is 
using coins as money, or talking of them as 
money, as things to buy other things with ; but 
not when any other business is on hand. 

' Twelve buttons are a dozen ' is true always, 
everywhere where people are speaking English, 
whatever use they mean to make of the buttons. 
But ' twelve pence are a shilling ' is not really 
true in itself; it is true as far as the business 
on hand is concerned, whenever people are 
talking about using coins to buy things with. 

You might go some day to a class in what 
is called physical science, where the business on 
hand is learning about the weights of things 







ARITHMETICAL SHORTHAND 55 

or the qualities of things; or to a class in 
cookery. If the teacher told you to go and 
fetch a dozen bottles, and you came back and 
said, ' Here are the twelve bottles you sent me 
for,' nothing would be said about it ; you would 
be all right. But if the teacher told you to 
take the weight of a shilling in some powder, 
or to take as much as would lie on a shilling, 
or to see what effect some liquid had when it 
touched a shilling, and he found you using 
twelve pennies instead, there would be some- 
thing said to you then! You would perhaps 
be told that you never could learn anything 
while you were so stupid and clumsy and 
inaccurate. 

I don't suppose you would ever make just 
that particular mistake. But I have known 
many children hopelessly puzzled over sums 
and other things ; and, when I came to see 
what puzzled them, I found they had been 
taking something some teacher had said, as 
meant to be true in itself, when it was only 
a shorthand sort of sentence which meant some- 
thing else : a shorthand which was true for 
one purpose but not for another. 

Try to understand always whether your 
teacher means what he says to be time always 
and everywhere ; or whether he means it as 
a bit of shorthand talk fit for that particular 



se 



LOGIC OF ARITHMETIC 






class. If you cannot find out for yourself, ask. 
Never go on using a sentence till you are sure 
whether it is meant for literal truth or con- 
venient shorthand. Think of the 1, the stroke 
which the old people cut on their tallies, and 
which meant, not one thing, but once putting 
all the fingers up and then down so as to begin 
counting again. Arithmetic is full, from begin- 
ning to end, of just such nice, clever, convenient 
bits of shorthand as that. Sums are difficult 
and puzzling chiefly because children forget 
this. 



KEEPING ACCOUNTS 

Suppose you have a shilling and spend three- 
pence, what have you left ? Ninepence. Is that 
all that is left ? 

Suppose that you go out with a shilling in 
your purse and spend three-pence on buying 
flower-roots, what do you bring back ? Nine- 
pence in your purse. Anything else? Any- 
thing not in the purse ? The roots. Well, is 
that an end of the matter ? What is to happen 
next ? The roots have to be planted, and then 
watered ; or else they will die, and you will 
have wasted your money. 



KEEPING ACCOUNTS 57 

You might have bought something else, not 
roots. But whatever you did with part of your 
money, something of some sort is left besides 
the change. If you bought food, it would be 
there ; and must be seen to that it might not be 
wasted. If you ate the food while you were 
out, you would bring back the strength you got 
out of it ; or if it was. unwholesome food, you 
might bring home ninepence and some very 
uncomfortable feelings. Whenever money has 
been spent, something remains besides the 
mere balance of cash. 

Yet if you have been taught to keep accounts, 
or if any grown-up person allows you to see 
how she keeps accounts, you know that when 
the money spent one week has been subtracted 
from the cash in hand, the page on which the 
things bought during the week have been 
written down is turned over, and only the 
remaining cash is entered on the new page for 
next week. 

Now shut your eyes for a minute or two, and 
think. See an account-book open. On one 
page is written : 

1 Mother gave me Is.' 

On the other page is written : 

• Plants, Sd.' 

You subtract the 3d. from the shilling, and 
write 9d Then you turn over the leaf, and 



58 



LOGIC OF ARITHMETIC 



on the next page you write, ' Brought over : $d.' 
The writing about the plants is hidden out of 
sight ; nothing is on the open page except the 
account of the money still left in your purse. 
The roots are not in sight ; they are in another 
room, waiting till you can go to them. But they 
have not gone out of the world : they are there, 
waiting ; and you will have to see to them 
presently. 

Then why is nothing written about them in 
the new page of the account-book? Think 
about that a little, and we will have a lesson 
about it next time. 



VI 



DIS-MEMBERLNG AND RE-COLLECTING 

The question you were to think over was : 
Why, when we turn over a page in an account- 
book, we enter on the next page nothing that 
was on the last page, except the balance in 
cash ; we make no mention of the things bought 
last week, though those things may still be left 
for us to deal with. The plants we bought last 
week still need watering ; part of the food we 
bought may be there still to eat ; the strength 
we got out of what we ate may remain to 
be used ; the illness we got if the food was 




DIS-MEMBERING, RE-COLLECTING 59 

unwholesome may be still uncured. Yet we 
make no mention of these things in this week's 
accounts. 

Why not? 

I thiuk you feel why not, though perhaps you 
cannot quite explain it. To-day we are going 
to have a little talk about that question. 

Human beings, you and I for instance, are 
finite creatures ; that is to say, we cannot be 
everywhere at once or do many things at once. 
Our bodies are made so that we can see only 
a short way across our big world ; and our minds 
are made so that we can only attend to a small 
part of any big business at once. All the rules 
of Arithmetic are made to help us to do big 
sums by attending to little bits at a time. If 
our minds were bigger we could do big sums, 
straight off, without any rules ; but, as it is, 
we have to attend to a bit at a time ; and the 
rules are made in order to fit the bits together 
properly. It is not only sums that have to be 
attended to a bit at a time, in proper order 
according to rule. Suppose you are going to 
a school-treat or picnic. If you were fairies, 
you could start off straight away ; and the 
proper clothes and boots would grow on you, 
if you needed them, as you flew along. But 
you are not fairies, so the clothes and boots 
have to be put on first, and carefully too, or 



60 



LOGIC OF ARITHMETIC 






you cannot go. Suppose you were thinking 
about what you would do out of doors, and not 
tying up your boot-laces properly, mother might 
say : ' Come, attend to your boot-laces now ; 
and when they are tied up safely, you can think 
of out of doors.' But ' out of doors ' has not 
gone away because you have to put it out of 
your mind for a little while and attend to your 
boot-laces. It is waiting, waiting, till you are 
at leisure to attend to it. So it is always. 
When we put anything out of sight and out of 
mind, in order to attend to something which 
for the moment we must attend to, the thing 
we put out of mind is not gone out of the 
world ; in some shape or form it is waiting, 
waiting, waiting, and will have to be reckoned 
with some time or other. 

Now we will do a little sum. 

How many days has a man lived who has 
lived forty-seven years ? 

If I asked you how many days a child has 
lived who has lived two weeks, you could tell 
me straight off, without any need to write on 
the black-board. There are how many days in 
a week? (7) and twice 7 are? (14). If your 
minds were giant minds, you could answer the 
other question, about the days in forty-seven 
years, just as easily as that one. But we, you 
and I, have such tiny, helpless little minds that 







DIS-MEMBERING, RE-COLLECTING 61 

we cannot manage a question like that. If we 
were giants we could pick pears off the top of 
a very big tree ; but human people have to 
climb up on ladders to reach high-growing 
pears. And in the same way we have to make 
a sort of mind-ladder before we can reach such 
big numbers as the days of forty -seven years. 
What are we going to do ? Write on the board. 
Yes ; tell me what I am to write. 

A 3 and a 6 and a 5. And then, under- 
neath, a 4 and a 7. 

What does the 3 stand for ? Three hundreds 
of days. And the 6 ? Six tens of days. The 5 ? 
Five single days. The 4 ? Four tens of years. 
And the 7 ? Seven years. 

Now tell me what I am to write. Seven times 
five are thirty-five . . . Why, what are you 
about ? I asked you to find the number of 
days in forty-seven whole years ; and here you 
are, telling me how many days seven times five 
days are. 

Had you forgotten the hundreds of days and 
the tens of days and the forty years ? Yes, for 
the moment ; we had to forget them, to push 
them out of our minds, so that we could attend 
properly to just the bit we were doing at a time. 
Go on then, seven times five are 35. There, 
I have written that down. Have we done now ? 
Is the sum right 1 Will you go home and tell 




62 LOGIC OF ARITHMETIC 

your mothers that you have found out at school 
how many days there are in forty-seveu years, 
and there are just thirty-five days in all those 
years ? Much good there would be in going to 
a school where that sort of thing was allowed 3 
Thirty-five is only the first rung of the ladder ; 
we haven't reached our answer yet. The three 
hundreds of days and the six tens of days and 
the forty years are waiting quietly, till we have 
finished attending to the seven times five ; next 
it will be the turn of seven times six tens to be 
attended to, while the hundreds are waiting, 
waiting. And now we multiply the three 
hundreds by seven ; and all this time the forty 
years are waiting, waiting, waiting, still. 

Well, now we have attended to each bit, and 
here are all our bits written down, one under 
the other '. But which of all the bits is the 
answer ? Which is the true right answer to 
the question we began about ? 

Alice, you say, 7x6 = 42, And, Mary, you 
think that 4x5 =20. And, Julia, your opinion 
is that 4 x 6 = 24. 

Which is the true view of the matter ? Are 
you going to quarrel about which is right ? Or 
would you rather have a nice long solemn argu- 
ment, each trying to prove that she has the true, 
the only true, answer to the question I asked ? 
1 See note at end of chapter. 




DIS-MEMBERING, RE-COLLECTING 63 

You won't do either? Well then, what will 
you do ? We must fit them all together to see 
what they all come to, before we have a right 
to tell people how many days there are in forty- 
seven years ; because our minds are so tiny, and 
the number of the days of the years are so very. 
very big. We have to make ourselves forget 
some things while we attend to other things ; 
but before we dare tell any one that we have 
found the actual truth, we must call back all 
that we made ourselves forget, and try to 
re-collect. 

Well then, 2 and 4 are 6 ; and 1 are seven. 
Is that right ? No ? Why not ? Some of the 
figures mean hundreds and some tens and some 
only single days, and we must sort them pro- 
perly before we fit them together, and arrange 
them so that each sort of figure shall stand for 
its true value. 

Now we understand better the question about 
the account-book. The use of the book is to 
keep account of money. Our minds are so small 
that we cannot think of cash accounts and other 
things at the same time. We wrote down 
once : ' Plants, 3d.,' because we may some day 
wish to look back and see what different things 
cost us ; but once writing it was enough : we do 
not wish to be constantly reminded of plants and 
other nice things, while we are busy adding up 



64 



LOGIC OF ARITHMETIC 






accounts ; we try to forget the things and attend 
to the cash accounts, just while we are using 
that book. But when we have done the accounts 
and shut up the books, we should recollect the 
things ; for everything that we have spent 
money on is still somewhere, in some shape, or 
form, waiting, waiting, waiting. 

And when we go to recollect the things, 
we must try to arrange them in their proper 
order so as to give to each its true value and 
meaning. 

That i8 enough for to-day. Sit slack, shut 
your eyes, and rest before you go to the next 
class. I am going to give you two mind- 
pictures : — 

A little boy was so excited about going to 
a treat that he would keep talking about it, 
and would not tie his shoe-laces properly. On 
liis way downstairs his lace came untied ; he 
stepped on it and fell and sprained his ankle, 
so he could not go to the treat after all. 

A servant was asked to get some children 
dressed to go out. She got them tidy, and their 
boots nicely cleaned and tied on ; the children 
felt impatient because they were in a hurry to 
go out ; but they knew they could not go till 
they were dressed, so they were good and quiet. 
When they were ready, they thought they were 
to go. But the nurse grumbled and said : 



WEIGHTS AND MEASUEES 65 

1 Now you are dressed and all tidy and clean, 
I don't want you to go out, for fear you should 
tumble your hair or get your boots dirty. I was 
told to dress you, and I have dressed you : that 
was the important thing ; that was my real duty. 
Going out is all nonsense. Sit still all the after- 
noon, and look at your nice clean boots.' 

Try to fancy how those children would feel, 
and what they would think. Try for a minute 
or two to fancy yourselves in the place of those 
children, and to think what you would feel like. 

Now open eyes and stand up. Go and get 
ready for the History Class. 

Note. 

The answers had better be written on the board at 
first in this form : — 



7 


x 5 


= 


35. 


7 


x6 


= 


42. 


7 


x 3 


= 


21. 


4 


x5 


= 


20. 


4 


x6 


= 


24. 


4 


x 3 


VII 


12. 


WEIGHTS 


AND 


MEASUEES 



On the Table : — A set of weights, one-ounce, two- 
ounce, four-ounce, half-a-pound, one-pound. 

What is this ? A weight. What weight ? 

■OOLX E 



Gfi 



LOGIC OF ARITHMETIC 



An ounce. What is it for? To weigh tilings 
with. Yes, this ounce weight is one of the 
things that has no proper use of its own ; its 
use is to make other things more useful by 
keeping them exact and orderly. 

Let us think of the uses of weights. They 
are used in shops, for the shopman to know 
exactly how much sugar or tea or meat he is to 
give for his customer's money. 

But they have another use as well. 

Very long ago people had no notion how to 
make cakes or puddings. When they caught 
an animal, they roasted it by a fire ; and when 
they found roots fit to eat, they roasted them 
b hot ashes ; and that was all they knew 
about cooking. Then they found out how to 
make pots of some kind, and then they found 
they could boil things in water. Gradually 
they found that some things improve the 
taste of other things, when they are boiled 
together : salt improves soup, and sugar im- 
proves puddings. But there must not be too 
much of one sort of flavouring. People began 
to try to find out how much of each kind of 
thing made the nicest mixture. They could 
find out only by trying. That is what we call 
experiment : a woman finds the soup too salt 
or not salt enough one day, she thinks it nicer 
when she puts in less salt or more salt ; then 




WEIGHTS AND MEASURES 67 

she tries to remember for the future what is 
the right quantity. 

But it is a pity for each one to have to make 
all the experiments for herself. It saves time 
and trouble, if one person's experiments can be 
made useful to other people. Suppose a woman 
is a very clever cook and finds out how to make 
tilings just right, it is well for her to be able to 
tell other women just what she found answered 
best. The other women's husbands and children 
may not have quite the same taste as the first 
woman's ; they may like a little more salt or 
sugar or spice than the first woman said ; but 
still, it is useful to know what other people find 
answer, and then we can alter it to suit our- 
selves and our families. 

Now how is the woman, who has made a nice 
cake, to tell others how much she put in ? . She 
has to use weights or else measures. She says: 
' I took a pound of flour, and a quarter of a 
pound of sugar, and half a tea-spoonful of baking 
powder,' and so on ; weights and measures are 
a language in which one person can tell others 
how much of anything to use. 

Well, a woman read in a book a receipt, as it 
is called, for making a nice small cake. One of 
the ingredients was an ounce of lard. But she 
was preparing for a tea for a great many people, 
and she wanted to mix all her dough at once 

E 2 



68 



LOGIC OF ARITHMETIC 



for several cakes. She wanted to mix sixteen 
times as much dough as the receipt told about. 
So she had to use more lard, sixteen times as 
much as the receipt told about. She had only 
one ounce weight. Look at my set of weights 
here ; there is no other of this size. Most sets 
of weights have only one of each size. The 
woman I am telling you of had only one weight 
of this size, and she wanted to weigh sixteen 
times as much. What did she do? Did she 
weigh out each ounce separately? If you think, 
you will see she could do something handier 
and quicker than that : she could weigh the 
whole sixteen at once with the pound weight. 

Now I should like you to go over the weights 
and see how they are arranged for use. 

If we want to weigh one ounce we use ? The 
ounce weight. If we want to weigh two ounces? 
The two-ounce weight. 

If we want to weigh three ounces, what shall 
we do ? There is no three-ounce weight. Put 
in the scales the two-ounce and the one- 
ounce. 

For four ounces? The quarter of a pound 
weight. 

For five ounces ? The quarter of a pound 
and the ounce. 

For six ounces ? The quarter of a pound and 
the two-ounce. 



The quarter and the two- 
ounce and one-ounce. 

For eight ounces ? The half-pound. 

(The teacher goes on thus up to thirty-one ounces.) 

So, you see, we can weigh any number of 
ounces we please, up to thirty-one ounces, by 
having only these five weights, one ounce, two 
ounces, quarter of a pound, half a pound, and 
one pound. 

If we want to weigh more than thirty-one 
ounces, we shall have to get other weights; 
such as two pounds, four pounds, &c. 

But we can weigh thirty-one different quan- 
tities with only five different weights. 



VIII 

MULTIPLYING BY MINUS 

Many persons who have learned a certain amount of 
Algebra are confused between what are called negative 
quantities, (movements, or actions), and the mental act of 
negation of quantity, (movement, or action) ; i. e. between 
the sign — which indicates active fact in the direction 
contrary to the one (arbitrarily) chosen as positive, and 
the Bymbol 0, which states denial of existence of fact. 
Practice should be given in exercises which involve these 
signs and show the distinction between them. 

It is also desirable, for many reasons, that children 
should be accustomed to use letters for unknown 



70 



LOGIC OF ARITHMETIC 



quantities at an early age. It assists the imagination 
in keeping a clear distinction between a quantity in 
itself and the effect on the quantity of a certain opera- 
tion or group of operations. Whatever a number may 
be, the group of operations represented by— 7 + 4 — 9+2 
diminishes it by 10. The statement # — 7 + 4—9 + 2 
= x— 10 has a meaning which can be made in- 
telligible at an early age ; whereas the mere statement 
— 7 + 4 — 9 + 2 = — 10 is confusing to a child's imagina- 
tion. Even for those who are never to loarn formal 
Algebra, the practice of using letters, for numbers which 
are to be the subjects of operation, imparts to Arithmetic 
something of Algebraic clearness of thought, which will 
be of use in itself; while those who are to learn Algebra 
should not be exposed to the mental violence of being 
introduced to Algebraic, notation for the first time, just 
when they will have to begin to learn the actual 
manipulation of unknown quantities by processes 
properly Algobraie. The following is a specimen of 
the kind of form in which elementary exercises in 
ordinary sums may with advantage be given. 

Black-board. 
T 



The figure on the board is supposed to be 
a counter in a shop. We are going to do a sura 
about buying something over the counter. What 



does the shopman call the thing in which lie 
keeps his money ? A till. Well then, we will 
call the amount of money in it T. What do we 
call the thing a customer carries money about 
in ? A purse. We will call the money in this 
customer's purse P. We do not know how 
much T is, nor how much P is : and we do not 
want to know just now. Provided that the 
customer has brought enough to pay for what 
she buys and the shopman has plenty of change, 
that is all we shall need to know at present. 
When we do not know a number, we often call 
it by the name of some letter. 

The customer sees boxes of sweets marked 
lid. She takes a box and pushes a shilling 
towards the shopman. What does he do ? Does 
he push anything towards her ? A penny. 

Does the shilling which she gives to him add 
to what is in the till or does it make it less ? 
Adds to it. When we add two things together, 
we put this mark between them +. So now 
we write what is in the till T-fL*. T means 
what was in the till before the customer came 
T has not altered. What is now in the 



in. 



till Is T + ls., which we read: T plus Is. But 
what has the shopman to give to the customer? 
Id. Is that added to T + Is., or taken from it ? 
Taken from it. We write that this way : — 1<Z., 
and we call it minus Id. 



72 



LOGIC OF ARITHMETIC 



So now this is how we write what is in the 
till; T + 1&-1A 

Just for to-day we are thinking about things 
as the shopman thinks about them ; we are 
agreeing with him. It is his till to which 
a shilling has been added, and his till from 
which a penny was taken. He thinks about 
the shilling as plus and the penny as minus. 
Some other day we will see what the customer 
thinks, but the shopman's thinkings are enough 
for to-day. He thinks pushing something from 
P to T is a plus action ; and pushing from T 
towards P is a minus action. As long as we 
are thinking like the shopman, everything that 
goes from P towards T is marked plus ; and 
everything that goes from T towards P is 
marked minus. 

For to-day, P to T is the plus direction. T to 
P is the minus direction. 

By such a lesson as the above, we lay a skeleton, 
round which may be grouped a groat variety of examples 
in ordinary addition and subtraction of money. The 
customer may ask for several articles in succession, and 
a bill be made out. Several customers may come in, 
and require amounts of change. All the money which 
passes should at iirst be registered as added to or taken 
from T. Sometimes the customer has the exact coins 
needed. In such case no change will pass ; this fact 
should be carefully recorded by writing —0. 

When the teacher wishes to extend the Bcope of the 
operations, he should go back to the one shilling given 
and one penny change ; so as to leave the children's 



MULTIPLYING BY MINUS 73 

thoughts free to understand the operation involved, 
undtstracted by questions of number. 

We are going now to look at the business of 
paying for things, from the customer's point 
of view, to see how she thinks about it. When 
she pushes a shilling towards the shopman, is it 
added to or taken from P? Taken from P. 
And the penny of change is? Added to P. Then 
the customer's account now is ? P — J.8. + 1(7. ; 
The shopman's is ? T + Is. — lcZ. 

After this, money sums should be done sometimes 
from the shopman's, sometimes from the customer's 
point of view. Occasionally, a sum should be written 
out from both points of view. Every shop transaction 
should be dealt with either as an addition to or a 
diminution of T or P ; or, preferably, as both. 

At a still later stage we suggest : — 

We are going to see whether the money has 
been paid and the change given all right. For 
that purpose we must know what was in the till 
before the customer came in. T was £3 9s. 6d. 
So on the shopman's side we ought to have 
£3 9s. 6d. + Is. -Id. P was £4 7s. 8d. 
Therefore the customer ought now to have 
£4: Is. &d,-ls.+ld. 

Then more sums are grouped rojind that conception. 
There is a further stage of tho same scheme. 

A customer asks for twelve articles value 
10s. each. She lays down 12 sovereigns. The 



74 LOGIC OF AKITHMETIC 

shopman hands back a shilling change for each 
sovereign : 12s. 

One of the articles is then found to be faulty. 
There are no more in stock. The customer says : 
' Never mind ; eleven will do for the present.' 
The shopman hands her back £1. Who has 
too much now ? Who must give a Is. in change? 
The customer. 

The account then stands thus : — 

On the shopman's side, 

T + £l2-12s.-£l + ls.; 
and on the customer's side, 

P-£l2 + 12s. + £l-l«. 

In order to emphasize this lesson, it should also be 
written out in another form : — 

JEL-U 

x(ldoz.-l) 



£12-125. 
-£l 

Now then, what does — 1 x — 1 come to ? In 
which direction must the last shilling be pushed? 
From P towards T. But that is, from the shop- 
man's point of view, the plus direction. So minus 
multiplied by minus comes to plus. 

Now from the customer's point of view : — 
* -£l + l*. 

x(ldoz-l) 

-£12 + 12s. 
+ £1-1*. 



ADDING MINUS 75 

This lesson carefully gone over, with the simple 
numbers given above, and afterwards repeated with 
more complex sums, would save all future trouble about 
the much-vexed problem of 'minus multiplied by 
jninus.' 



IX 

ADDING MINUS 

Black-board. 
S H 



M M M 

S is a shop where a boy works. H- is the 
house where he lives and sleeps. C is a house 
where a customer of his master lives. Each M 
is a mile-stone on the road. The distances 
from S to H and from M toM are, each, a 
small picture of a mile along the road. 

Now, do you remember what those letters 
stand for ? Because I don't want you to have 
to think about that presently when I am ex- 
plaining something else. So say those letters 
over : S, shop ; H, boy's home ; C, customer's 
house ; M, mile-stone. Again : S, shop ; H, 
boy's home ; C, customer's house ; M, mile- 
stone. 

You see which way ;the arrow points. Do 
you remember that we Jiad a lesson about 



76 LOGIC OP ARITHMETIC 

money pushed across a counter? When it was 
pushed across one way, it added to the shop- 
man's money ; so, when we were doing his 
accounts, we called that direction plus ; and 
when he pushed change back, we called that 
minus. Miles on a road are sometimes counted 
in the same way : we call one direction plus 
and the other minus. Well, the boy who works 
at the shop lives one mile from the shop in the 
direction that we are going to call plus ; what 
then shall we call a mile in the opposite 
direction ? A mile minus ; he walks a mile 
minus in going to work ; every evening he 
walks a mile plus in going home. C is two 
miles further on than H. So all the distance 
from S to C is how many miles ? In what 
direction ? S to C is three miles plus. 

One night the master says : ' Bill, there is 
a parcel to be taken to Mr. Smith's house (C). 
You live in that direction ; you must take it.' 

What then has Bill got to do? How many 
miles plus has he to walk ? Three ; and what 
has he to do next ? He must get home some- 
how. What must he do? Walk two miles 
minus. So three miles plus and two miles minus 
brings him to the same place, H, as he gets to 
on other evenings by walking one mile plus. 

We said that the boy's task, set him by his 
master, was to walk three miles plus. He 






neutralized or counteracted part of that walk 
by adding to it a walk of two miles minus. 

But now suppose that before Bill starts with 
his parcel a workman says to the master : ' I 
am going to see a friend who lives close to 
Mr. Smith's house ; may I take the parcel, sir?' 
And suppose the master gives leave. What 
has the workman done to Bill's task of three 
miles plus ? He has taken off from it two 
miles plus. Bill still has to walk one mile plus, 
because he must get home. 

If the workman takes the parcel, he subtracts 
or takes off from Bill's task, two miles plus ; if 
Bill takes the parcel he neutralizes two of the 
plus miles by adding to his three miles plus 
another walk of two miles minus. His walk 
in either case ends at the same point H. But 
now I want to know : — are the two ways of 
getting home the same in any other respect ? 
For instance, will he get home to supper at the 
same time ? No, he will be more than an hour 
later for supper if he takes the parcel himself. 
And what about exercise and health and all 
that ? If Bill is cashier, and has been sitting 
still all day in a close shop, the five miles walk, 
three plus and two minus, would do him a great 
deal of good. But if Bill is a delicate little 
chap and has been on his feet all day, that five 
mile walk instead of one mile might make him 



78 



LOGIC OP ARITHMETIC 



quite ill. It would make a difference to wear 
and tear of shoe-leather, and to the length of 
time he would have for playing with his sister 
before going to bed. The only thing about 
which adding two miles minus comes to the same 
as subtracting two miles plus, is the point, one 
mile from the shop, where he gets to at the end 
of his walk. 

So you see, if you are asked a question about 
the point which Bill will reach at last, it will 
be quite right to say, ' Adding minus two comes 
to the same as subtracting plus two.' If you 
are asked about Bill's health, or bis shoes, or 
the .time he gets his supper, you must not say 
anything of the kind ; because, in relation to 
those questions, it would not be true. 

Mind-Picture. 

Bill trudging home tired ; near the middle M ; 
with his back to C and his face towards H. He 
is looking tired, and is glad to have got as far 
as M on his way back. 



DIVIDING AND SHARING 

What is the half of five ? Two and a half. 
If two children are to share five ounces of any- 
thing, what is the share of each ? Two and 




a half ounces. Is that so always ? Are you 
sure ? Well now, let us see. 

Suppose two children are to share a cake 
weighing five ounces, the share of each is ? 
Two and a half ounces of cake. How do we 
divide it \ Do we give one all the top and the 
other all the bottom ? No, that would not be 
a fair division ; we cut it so that each gets half 
the currants, half the sugar on the top, and 
half the bit of candied peel in the middle, as 
well as half the dough. 

Suppose five marbles, weighing an ounce each, 
are given to two boys to play with, what is the 
share of each % Two marbles ; and there will 
be one over (as you say in division sums). What 
would the boys do with the one over ? I think 
they would put it on the ground between them, 
and each shoot at it in turns, with the marbles 
that were his own share. 

Suppose I have a doll weighing five ounces, 
and give it to two little girls. What is the 
share of each ? Will they cut it in two ; and 
each have half a dolly ? Shall each have one 
leg and one arm and half a head ; half the 
calico, half the plaster, and half the saw-dust ? 
No. Well, what would be the share of each ? 
If it was a naked doll, there would be its clothes 
to make ; they would do the work between 
them. When they had time to nurse it for an 



80 



LOGIC OF ARITHMETIC 



hour, they would each nurse it for half an hour. 
When they had it to tea, they would sit one 
each side of it. That would be the fair sharing 
of a five-ounce dolly. 

Now suppose a flower-bulb, weighing five 
ounces, is given to two children, how would 
they share it 1 Would they cut it in two down 
the middle ? No ; they would plant it, and 
take turns at watering and tending >it. 

Children who have become mechaniealized, here add : 
4 And share the flowers when it blossomed.' 

Do you mean what you are saying *? Just 
think a minute. Babies might cut a hyacinth 
up for the sake of sharing the flower-bells 
between them, and each being able to say, 
' These are mine ' ; but I never saw two children 
old enough to tend plants who would do such a 
stupid thing. What would you really do if the 
plant were yours? Keep it growing, to look 
at, and show to friends ; and how about the 
sharing? If one child looks at it, does that 
prevent the other seeing it ? If one smells it, 
does that prevent the other smelling it ? If one 
child brings in her special school-friend to see 
it, does that prevent the other from bringing 
in her special friend to see it too ? No. But 
neither child can say : 'This is my plant.' 

So now this is what we have come to. 

If two people are to share five ounces of 



IN WHAT CONSISTS ECONOMY 81 



cake, each is to have two and a half ounces 
of cake, each share to contain half of each kind 
of ingredients that the cake is made of. 

If they are to share five marbles weighing 
an ounce each, the share of each is two marbles 
and the fun of shooting them at the fifth marble. 

If they are to share a five-ounce doll, the 
share of each is half the work of dressing it, 
the fun of nursing it for half the time they 
have to spare ; and the fact of sitting beside it 
at tea just as if it were all her own. 

If they are to share a flower-bulb, the share 
of each is, half the work of tending it, all the 
pleasure of seeing it, all the pleasure of smelling 
it, all the pleasure of letting friends see it ; and 
none at all of the pleasure of saying : - This is 
mine, my very own, and no one may look at 
it or smell it without my leave.' 



XI 

IN WHAT CONSISTS ECONOMY 

Black-board. 
A. # B. 

Things such that I Things such that 
no one else can • other people can 
have the part I have the part 
thereof that I • thereof that I 
have. I have. 

BMLU F 



82 



LOGIC OF ARITHMETIC 



You remember what we said about sharing 
cake and sharing a lily-bulb. If two children 
share five ounces of cake equally, the share 
of each is ? Two and a half ounces. If one 
has three ounces, the other can only have ? 
Two. If one has four ounces, the other can only 
have ? One ounce. If one has five ounces, the 
other gets ? None. Now look at the black- 
board. On which side shall we write C&ke? 
Under A. 

Now about the lily-bulb. If two children 
share a bulb, the share of each is ? — 

Half the labour of growing it. 

All the pleasure of seeing it grow. 

All the pleasure of smelling the flowers when 
they come out. 

All the pleasure of seeing friends enjoy the 
flowers. 

None at all of the pleasure of saying : ' This 
is my very own, and no one else can share it,' 
So we shall write the bulb under? B. 

Suppose two people grow bulbs for sale, they 
get ? Money. What happens to the money ? 
It is shared between them. If they get six- 
pence for a bulb, the share of each is ? Three- 
pence. If one has four-pence, the other can 
only get ? Two-pence. We will write money 

got by selling bulbs under ? A. 

Now I want you to think about a little row 




IN WHAT CONSISTS ECONOMY 83 

of houses, six comfortable houses, all about the 
same size, built near together on a common, 
away from shops. The gardens are as yet 
scarcely planted at all. Suppose one of you 
is the mistress of the end house of the row. 

One morning, two hawkers come up the road 
and come past your house first. One has on 
his barrow a lot of nice fresh fruit, about as 
much as he thinks the people in your row will 
want to buy. The other has young plants. 

Now remember, we are not here talking of 
shipwrecked sailors, or desert islands. I am 
not asking what people should do who are shut 
up together, with not enough food. I am not 
asking what is the heroic thing to do in ex- 
ceptional circumstances ; but of what is the 
common -sense thing to do *in commonplace 
circumstances. There is enough fruit on the 
barrow for every one in the row; and every one 
has enough money to buy some. We will 
admit at starting that the first duty of a house- 
keeper, in ordinary civilized society, is to see 
that there is proper food in the house for all 
the household ; so the first thing you do, when 
the hawkers call, is to buy as much fruit as 
your household will need while it keeps good. 

The people from one of the other houses are 
coming in to tea ; so you buy enough for them 
to have plenty as well as your own family. 

p 2 




84 LOGIC OF ARITHMETIC 

We will suppose that all the rest of the 
necessary provisions are in the house already. 
You have just bought fruit enough ; and you 
have a few shillings which you decide to spend 
on 'extras.' There are two things you can do 
with them. 

It comes into your head that it would look 
rather grand if your neighbour were to see 
on the table a huge heap of strawberries : — 
two or three times as many as could be eaten 
that day. Will you buy that extra fruit ? 
Or will you buy some plants for the garden ? 

Let us see what would come of each of these 
proceedings. 

Suppose you spend your spare shillings on 
buying more fruit than you need, some other 
family will not *be able to get any. So we 
must enter strawberries on the A side of 
the black-board. 

If you buy up that fruit, what happens to 
it ? A good deal is left for next day ; not 
quite as nice as it was the first day. No one 
can enjoy it now quite as much as the people 
who got none would have done if they had 
got it while it was fresh. Everybody in the 
house is perhaps tempted to eat a little more 
than they otherwise would do, because it is 
there and must not be wasted. Yet in spite 
of that some of it is wasted; goes really bad 




EN WHAT CONSISTS. ECONOMY 85 

and has to be destroyed at last. Meantime 
there has been about the house a faint smell 
of not quite fresh fruit, which takes away 
everybody's appetite and makes every one un- 
comfortable. For the next few days, no one 
cares as much about fruit as they did before. 

You will often find this the case about things 
in the A class of this division, (point to black- 
bird.) It is right that you should have enough 
of each of them ; but the least bit more than 
enough is too much, too much even for your 
own good. 

Well now, suppose that instead of buying 
too much fruit you buy plants for your garden, 
what happens ? All the people in the house 
enjoy them ; visitors enjoy them ; and the 
people who pass by on the road enjoy them 
too. 

We will write the plants down in Column ? 
B. 

I have known many servants who liked 
living in houses where the mistress bought 
what is called in Ireland ' lashings and leav- 
ings ' of all sorts of food-things ; they called 
a mistress stingy who bought only as much as 
was of real use. You might see, in such houses, 
ends of joints going bad, bits of loaves on the 
floor amoDg dirty boots, milk going sour, fish 
going stale, candles guttering away in waste ; 



86 



LOGIC OF AKITHMETIC 



and the garden quite neglected because the 
family were too poor to keep it up and had no 
money to spend on plants. And the servants 
thought such masters 'rale gintry.' You all 
know that that was because these servants 
were ignorant and uneducated. I think you 
will find that this is a good test of true 
education. 

(The teacher now rubs out the instances, leaving only 
the headings of the two columns ; and writes under 
them tins sentence : — ) 

True education tends to make 
people satisfied with just enough 
of the things in Column A, and 
leads them to spend spare time, 
money, and energy on things in 
Column B. 

Read what is on the black-board. Copy it 
in your books. Read it out. Shut your eyes 
and try to say it. Learn it by heart (in 
preparation time). 

Now I am going to tell you something rather 
difficult. You will not quite understand it yet, 
but if you think about it and fix it in your 
minds you will understand it some day, and 
it will help you to understand many other 
things. 

You might, if you could afford it, buy more 
kinds of plants than you could attend to 




properly, or than the garden could well hold. 
Some people do that kind of thing ; they can- 
not make up their minds at once to do without 
what they cannot use ; so they crowd up their 
houses and premises with things that are in 
the way. 

Shut your eyes and think of this for a 
minute or two .—Plants that are crowding 
each other up, and preventing each other glow- 
ing, or that are not planted and are littering 
up the house, are not in Class B ; I am not 
sure that they get as far as to be properly in 
Class A. 



XII 
ECONOMY OF MIND-FORCE 



True education tends to make people satisfied 
with just enough of the things in Column A, 
and leads them to spend spare time, money, and 
energy on things in Column B. 

Bl^ACK-BOARD. 

A. B. 

Things such that • Things such that 
no one else can I other people can 
have the part that • have the part that 
I have. * I have. 



88 



LOGIC OF AEITHMETIC 



You know that when I want to make you 
remember a sentence which I have written on 
the board, I let you all say it at once. But 
when I want to help you to understand some- 
thing, or to see whether you understand, I allow 
only one child to answer at a time. I will 

write: — repeating a thing to help me 

to remember, in Column ? B. 

I will write: — showing how much I 
know and how far I understand, in 

Column ? A. 

There are several things you have to learn in 
school, besides the actual lessons in the books, 
such as sums and geography. One of the most 
important is to answer questions properly. You 
must learn to think what was the exact question 
asked ; to think what the question means ; to 
answer it quickly, quietly, and politely ; to tell 
the truth, the whole truth so far as you know 
it, and nothing but the truth, about that 
question ; to put it in plain words, and not to 
use unnecessary and roundabout phrases. All 
that needs practice. It is the teacher's duty to 
give each child a share of such practice— just 
as it is the housewife's business to give every 
child a proper amount of food. And as we said, 
while Mary is answering, Claire cannot answer. 

Therefore we put: — practice in answer- 
ing questions, in Column ? A. 



child 



ECONOMY OF MIND-FORCE 89 

Another thing that is necessary to learn is to 
listen to other people's conversation, without 
interrupting till your turn comes. If Alice is 
answering me, and Mary is sitting quiet, can 
Claire sit quiet too? Yes. Then we will 

write: — practice in listening to con- 
versation without interrupting till 

my turn comes, in Column? B. 

But we must not sit idle while other people 
are discussing interesting or useful things, we 
must learn to profit by what is said. If I am 
teaching Alice and making her answer, and 
Mary is listening and learning by what Alice 
and I say, can Claire listen and learn too ? Yes. 

Then we will write : — learning from what 

is going On, in Column ? B. 

It is necessary to collect a good stock of mind- 
pictures for our future use. While one child is 
sitting nice and quiet making a mind-picture, 
does that hinder the others from doing so too ? 
No; the quieter each one sits, and the more 
steadily she makes her niind-picture, the better 
every one else can do the same. We will write : — 

making mind-pictures, in Column ? B. 

It is necessary that a child should not pass 
up into a higher class (form, or standard) till 
he is quite fit for it. Therefore, there must be 
pass-examinations, to see who is fit. If one 
child passes an examination creditably, does 



90 



LOGIC OF ARITHMETIC 







that hinder another from doing so ? No. We 

will write :— passing school examina- 
tions creditably, m Column ? B. 

It is necessary that Governments should know 
who is fit to become a doctor, or a lawyer, or 
a school-teacher, or a Civil Service clerk, or 
a postman, because they ought not to allow 
unfit persons to undertake responsible posts. 
If one man shows he knows his work properly, 
does that hinder another from showing that he 
knows it too ? No. We will write : — passing 

professional pass-examinations, under 

Column ? B. 

But for some purposes it is right that the very 
fittest persons who can be got should be chosen 
for a post. It is sometimes right therefore that 
the Government should know, not only who is 
lit, but who is most fit. If one man is at the 
top, can another be at the top ? No. We will 
write . — being at the top, in Column ? A. 

Now let us read over carefully our two 
columns. 

(Bead out the columns.) 

Now I am going to speak about something 
which puzzles many children. You are always 
being told that it is kind to help other people 
and share with them, yet if you help another 
child at an examination you are told it is 
naughty ; and if you let another child help you 



you are punished. Even at class, if I ask one 
child a question, and another whispers the 
answer to her, or writes it for her to see, I 
stop you at once. 

Let us see what all that means. 

If I say to one child, ' I do not know whether 
any dogwood grows in this neighbourhood ; will 
you keep a look-out when you are out walking ? ' 
Would it be right for another child to help her 
to look for it ? Yes ; because the thing I want 
to know is whether there is dogwood ; and the 
thing the child wants to do for me is to find 
the dogwood if it is there. Two pair of eyes 
are better than one ; two children, by helping 
each other, are more likely to succeed in doing 
what is wanted than one alone. 

But if I ask one of you a question at class — for 
instance, what are nine times eight ? — I do not 
want to know what nine times eight are ; I know 
that already. What I want is, first, to know 
whether that child remembers nine times eight 
and can say it without help ; and, secondly, to 
give that child practice in grasping for herself 
the question she was asked and answering it 
For herself. If another child interferes or tries 
to help, she does not help, she hinders the very 
purpose for which the question was asked. 

So it is at examinations. The examiner does 
not want to know the answer to the questions 



4 



92 



LOGIC OF AEITHMETIC 



he has asked ; he wants to know which children 
are fit to begin the work of a higher class. If 
you ' help ' a friend (as you call it) to answer 
questions, you are really hindering her progress, 
by helping to get her into a class she is not fit 
for. You help her to be set to lessons too 
difficult for her. You help her to have duties 
which she cannot properly do. You help her 
to brain-muddle and overstrain and many sorts 
of bad things. 

Black-board. 

When the object of doing a thing 
is to get the thing done, it is right to 
accept the help of other people. 

When the object of doing a thing 
is either to get practice in doing it, 
or to show that I can do it, I must 
not accept help from other people. 

There is something that I should like to tell 
the elder children just here. Little ones must 
please sit still, and practise trying to learn 
what they can from conversation between their 
elders. 

There are, in every subject, some parts which 
can be learned on purpose to show that you 
know them ; for instance, on purpose to pass 
examinations in them. There are other parts 
which cannot be learned so. which must be 







learned while you are not thinking of showing 
what you know. Now you can never really 
thoroughly understand either of these parts 
unless you also learn something about the 
other. You want to know why this is so? 
Well, I will answer that question after you 
have answered me a few questions. Which 
makes a man strongest for exercise, walking 
all day, or leaving off to take food sometimes ? 
Which has the better nourished body, the boy 
who sits trying to stuff down food all day, 
or the boy who does a fair amount of running 
about between times ? Which will put forth 
most leaves and fruit to show, the plant that 
has been cut from its root and now has only 
branches, or the plant that is growing from 
a root hidden under ground ? 

I once knew a seliish, greedy little boy, who 
would rather make himself ill than let the 
servant (who made the pudding, and brought 
it to him, and was going to wash up his plate) 
have any herself. Do you think that boy 
could learn the true use of food, and the proper 
care of his digestion, while he thought of his 
food in that sort of way ? You know that 
be could not learn so. Why ? You cannot 
explain, but you know he could not. 

Well, tell me why all those things are so, and 
then I will tell you why you understand your 



94 



LOGIC OF ARITHMETIC 



examination work best on the whole, if you do 
some portions of work which do not tell at 
examinations at all. It is our ' nature to,' as 
the old rhyme says. We are made so, and we 
cannot help ourselves. 

All the great men whose names live for 
centuries after their death, did a great deal 
of work which the world never heard of. Men 
who will do nothing except that part which 
they can show, never do anything which lasts. 

A minute or two of silence. 

Now all of you read aloud what is on the 
black-board. Copy it into your books. Read 
it again. Shut your eyes. Try if you can say 
it. Learn it by heart (in preparation), and say it 
to me next time. 

I am not going to give you a mind-picture 
to-day. Shut your eyes and sit at ease, and 
try to put together in your minds the different 
things we have been talking about. 

Five minutes' silence. 



XIH 

EXERCISE IN RELEVANCE; INTRO- 
DUCING IDEA OF PROBABILITY 

Suppose a pony is shut up in a field alone. 
How many heads will be in the field ? How 



many legs ? Eyes ? Hoofs ? Tails ? Hands ? 
How many pieces of mischief do you think will 
be done in an hour ? Suppose another pony is 
turned in. (Repeat same questions.) Twenty ponies 
come in. (Eepeut.) A hen comes in. (Repeat.) A boy 
comes in. (Repeat.) Another boy comes in, the 
second boy has a monkey. The master comes 

in. (Repeat the questions each time.) Now let US put 

all that on the black -board. 










3£ 


3 


09 


00 •S's 


ca 


60 9 2 & 


§ g-S 


9 

m 


h3 W H Eh 


wSS 


1st Pony 1 


4 2 4 1 





2nd Pony 1 


4 2 4 1 





20 Ponies 20 


80 40 80 20 





Hen 1 


2 2 1 





Boy 1 


2 2 


2 


2nd Boy 1 


2 2 


2 


Monkey 1 


4 2 1 


4 


Master 1 


2 2 


2 



(It is important that all zeros in this exercise be 
filled in. For though zero is not anything, negation is 
a mental act, the cognition of a fact. It is important 
that children should learn to cognize the fact and deal 
rightly w th it, at an early age. 

The. seventh column is to be filled in, in each case, 
according to the decision of the class after discussion of 
the probabilities of that case. 

Exercises such as the above can be varied indefinitely. 




LOGIC OF ARITHMETIC 



They are of great importance ; they fix attention on the 
questions of relevance, and of cumulation or neutraliza- 
tion (e. g. the advent of the master adds to the number 
of legs, is irrelevant to questions about hoofs or tails, 
and diminishes the amount of mischief). They call 
attention to the line of demarcation between certain, 
calculable knowledge (number of heads, &c), and pro- 
bable knowledge, or that which is contingent on facts 
not yet known, or not ascertainable (the character of 
the boys). They also call attention to the fact that an 
additional element may either add its own bulk to that 
previously existing (heads, tails), or may raise that 
previously existing to a higher power. A boy once 
suggested to me that, if two boys were in a field, they 
would not only do mischief themselves but would 
start the ponies trying to break fences ; and added : — 
' One boy alone might not think of it, but two together 
would be bound to.' I affirm tliat the boy's mathe- 
matical insight must have been increased by his having 
mado such a suggestion. All such ideas are better 
suggested by examples of the above kind, where no 
intellectual work is involved in the question itself and 
the mind is free for now conceptions to surge up of 
themselves from the ' abysmal depths of consciousness,' 
than by exp]anations thrust in ab extra, while the child's 
mind is occupied in struggling with the difficulties of 
a problem in Aritlimetic or Algebra. 

For town children, playground may be substituted 
for field, dogs and cats for ponies). 

MlND-PlCTUBE, 

Shut your eyes, &e. See a field ; see ponies, two 
boys, arid monkey. Monkey jumps on the back of 
a pony and frightens it. Boys shout and scamper. 
Ponies become wild and break fences. Hen runs 
away scared and falls into pond, &c, &c, &c. 

Master comes in and restores order. 



XIV 
EXERCISE ON ZERO 



Shut eyes, &c. Make a mind-picture : — Me 
lifting the chalk to the black-board. I make 
one stroke and then put my hand down. I do 
this action three times ; how many strokes will 
be on the board ? 

If, instead of making one stroke on the board, 
I made two and put my hand down ; how many 
strokes would be on the board when I had done 
the action once ? Twice ? Three times % Four 
times ? Before I had done it at all ? 

Open eyes, sit up. Shut eyes, &c. Make a 
mind-picture : — A clean black-board, me holding 
the chalk aud then putting it down, without 
touching the board ; what would be on the 
board? Nothing. Now make another picture : — 
Me making a stroke. Now I rub the stroke 
out. What is on the board that you now see 
in your mind ? Nothing. So if I do nothing, 
or if I make a stroke and rub it out, the result 
is the same as far as the board is concerned. 
Were the two ways of getting it the same? No. 

That was a mind-picture black-board. Now 
open eyes and look at the real one. Here is 
a quite clean board; I make a stroke ; I rub it 
out. Is it really quite as clean as when it has 

BOOLI Q 



98 



LOGIC OF ARITHMETIC 



been cleaned on purpose for class ? No. What 
do you see on it ? A smudge. 

Suppose I lifted the chalk to the board, and 
made a dot and no stroke at all ; how many 
strokes would be on the board when I had 
done the action three times ? Four times ? Six 
times ? Nine times ? Once ? Before I had 
done it ? 

One stroke no time ? 

Two strokes no time ? 

No stroke three times? 

No stroke one time ? 

No stroke no time ? 



XV 



1 = oo 



The idea of fraction is readily introduced by accus- 
toming children, when the concept 'child' is taken as 
arithmetical unit, to think of that unit as divided into 
lialves, each side being a half; or the unit may be 
a monkey, each hand representing a quarter. For 
instance, a common exercise in Arithmetic is the 
following : — 

If each boy is to have two apples, how many 
shall I want for two boys, three boys, four boys ? 
&c. 

The problem might be stated thus : — 

If each boy is to have an apple in each hand, 
how many will be given to two boys ? Three 




1 _ 



= CO 



99 



boys ? &.o. One boy ? Half a boy ? If each 
monkey is to have two nuts in each hand, how 
many will three monkeys have ? Two monkeys? 
One monkey ? Half a monkey ? A quarter of 
a monkey ? Three quarters of a monkey ? 

Such questions as tho above may son mi foolish, 
because teachers are not yet accustomed to see their 
importance in the development of arithmetical faculty. 
Let us once learn to think of the human mind as 
intended to build up the material sciences round an 
organic skeleton made of acts of positing the unit, 
negation, fraction and reconstruction of the broken unit, 
and our estimate of the relative values of many things 
in education will undergo rapid change. Every ele- 
mentary exercise in number should be applied to the 
concepts unity, zero, one-half, one-quarter, &c. In con- 
structing his tables, the child should bo asked ' Twice 
one ? ' ' Twice nought ? ' and ' Twice a half ? ' as often 
as he is asked ; Twice three ? ' 

Another useful exercise is the following : — 

I have twenty apples on a plate. Boys pass 
through the room, each coming up for his 
allowance. If I give one to each boy, how 
many bbys can I serve before my stock is 
exhausted ? If I give one to each half-boy ? 
If I give two to each boy ? If I give two to 
each half-boy ? If I give four, five, ten, twenty, 
to each boy ? 

Then come down the series. 

If I give ten, five, four, two, one, to each boy ? 
If I give half &xi apple to each boy ? Half an apple 
to each half- boy, a quarter-apple to each boy ? 

a 2 



100 



LOGIC OF ARITHMETIC 



The questions should be kept playing round the 
fraction of apple and fraction of child, till the children 
see quite clearly through the whole process, and are 
familiar with the conception that, at the point where 
the share of each child ie one apple, L e. where the 
unitary concept of the class 'children' coincides with the 
unitary concept of the class ' apples,' a crisis is reached ; 
something happens ; at that point the number of boys 
is equal to the number of apples ; on one side of it the 
number of boys is less than that of apples ; on the 
other side it is greater. 

The series of questions should be repeated with ten 
apples in the plate ; then with forty apples, then the 
three series (ten, twenty, forty apples) should be inter- 
mixed. And as soon as it can be done without 
confusion, this question should be asked : — 

If I give none to each boy, how many can 
pass through the room before my stock is 
exhausted ? 

We thus introduce the true mathematical conception 
of ' infinity,' free from all that is hazy or doubtful, or 
which makes a strain on the nerves or imagination ; we 
call attention to the simple fact that no number of boys 
passing through the room affeets the remaining stock of 
apples in any way ; that, when the share of each boy 
passes from fraction of apple to negation of apple, the 
relevance of the number of apples on the plate to the 
number of boys is suddenly broken. 

Then ask : — 

If the share of each half-boy is none, what is 
the share of each boy ? And how many boys 
can pass through before the supply is exhausted? 

Exercises of this kind should be strictly limited to 
such simple fractions as are indicated by the two hands 
of the child and the four hands of the monkey. No 






attempt should be made to use them to impart any 
premature information about the branch of Arithmetic 
commonly called ' fractions.' 

The object of these examples is not to teach 'fractions,' 
but to supply elementary Arithmetic with that normal 
thinking-fibre which is, for a human creature, essential 
to clear ideas about anything, and which is generated, 
as above stated, by the mutual inter-action of the ideas 
of unit, negation and fraction. 



XVI" 
EQUIVALENT FRACTIONS 

Sit at ease, shut eyes. Make a mind- 
picture. 

Each one of you is to think of some grown- 
up person that he likes. I am going to call 
the person ' Mother ' ; but you may think of 
Father or Auntie or any one else you like. 
Now then. 

There is a plate on the table, and a knife, 
and an apple. Mother cuts the apple into two 
pieces the same size. We call those pieces ? 
Halves. Now Mother cuts each of those pieces 
into two. We call those pieces halves of 
halves, or? Quarters. Mother eats one of the 
quarters, and says you may have the rest. 
How many quarters are left for you ? Three. 
Two of them are the two halves of one half- 



102 



LOGIC OF ARITHMETIC 



apple ; the other quarter is one of the halves 
of the other half-apple. 

Now, what is left for you when some one 
takes one quarter of an apple and leaves you 
the rest ? Three quarters. 

Open eyes, sit up. 

Black-board. 

If a person takes one quarter of 
anything and leaves me the rest, 
what I ge$ is three quarters. 

Is that right ? Do you quite understand it ? 
Now write it in your table-books. Head it 
out. 

Now sit at ease again. Shut eyes. Make a 
picture. Me with four of you children. Each 
of you is to think of himself and three others ; 
any three others you like. 

There is a table in front of me, and three 
apples on a plate. I am going to divide those 
three apples among the four children. If I give 
one of the apples to each of- your friends, what 
will be left for you ? Nothing. Well, that 
won't do, will it? 

We must try a better way than that. I cut 
an apple into quarters. How many pieces will 
there be? Four. I give one piece to each of 
you ; what sized piece will each child have ? 
A quarter. What will be left of that apple on 






the plate ? Nothing. Then I take the second 
apple and give each child a quarter of it. So 
now each child will have ? Two quarters of 
apple ; and there will be left on the plate one 
apple and no piece. Now 1 cut up the third 
apple and give each child a quarter. And now 
there is left on the plate ? Nothing. So we 
have shared up the three apples among four 
children. And the children have all got just 
the same quantity. What is each child's 
share ? Three quarters of apple. Are you 
sure you understand ? Shall we go over it all 
again ? 

(Repeat the paragraph if uecessary.) 

Open eyes. Sit up. 

Black-board. 

If three apples are shared between 
four children, the share of each is 
three quarters of an apple. 

Copy into your table-books under the 
sentence you wrote last. Read both sentences 
over. You see that if three things are shared 
between four children, each one's share is the 
same size as it would have been if there were 
only one thing, and some one took away one 
quarter and left him all the rest. What is 
alike in the two cases ? Your share. Only 
that. Everything else is different. When we 



104 



LOGIC OF ARITHMETIC 



are talking short, we say that ' A quarter of 
three is equal to, or the same as, three quarters 
of one.' And we write it this way : — 

Black-board. 
|ofl = iof3. 

Copy that in your books. 

When this exercise is gone over, another day, the 
following should be given as a variation :- — 

I cut up the three apples into quarters. 
How many pieces axe now on the plate ? 
Twelve. And there are four children. How 
many pieces can I give to each child ? Three. 
And each piece is ? A quarter of apple. So 
each child's share is ? Three quarters of apple. 

We now have this : — 

Black-board. 

Three quarters of one apple. 
A quarter of each of three apples. 
A fourth part of twelve quarter- 
apples. 
All are the same size. 

Copy that into your books. 

If four children are to share one apple, how 
many quarter-apples does each child get ? 

If two children are to share two apples, how 
many quarter-apples does each get ? 




Before children are introduced to the practice of 
' cancelling ' out, in proportion or fraction sums, it 
would be desirable to introduce them to the idea under- 
lying such processes in some such way as this : — 

Suppose I have four apples to share among 
eight boys, there are two ways in which I can 
do it. I can cut up all the apples into halves, 
and give one piece to each boy, or I can give 
a whole apple to each: two boys, and let one of 
each pair divide the apple belonging to that 
pair. I have, to begin with, eight boys and 
four whole apples on a plate. I shall have at 
the end eight boys each holding a half-apple. 
One of the ways of doing this is to make first, 
four groups, each consisting of two boys and an 
apple. The other way is to cut up all the 
apples into halves, and give one to each boy. 
Whichever way I set about it, I get at the last 
eight boys each holding half an apple. 

It is desirable to avoid suggesting to the child's 
imagi nation, at any stage of the process, that two boys 
and an apple is supposed to be in any sense the equiva- 
lent of either eight boys and four apples, or one boy 
and half an apple. Therefore make a clear mind- 



106 



LOGIC OF ARITHMETIC 



picture of the four groups, each consisting of two boys, 
one of whom is engaged in cutting an apple into 
halves. 

Enter in hooks (pupil's own part) : — 

' If four apples are to be shared among eight 
boys, the share of each boy is the same, whether 
one man cuts up all the apples, or whether four 
people each cut one apple in halves.' 

All lessons concerning equi valence «f fractions and 
the dividing of numerator and denominator by the same 
factor should be linked in the children's minds with 
this fundamental conception of a set of molecules or 
groups, each consisting of so many boy-units and so 
many apple-units ; it should be shown that what is can- 
cel U'll out is the number of such groups ; and that the 
number of the groups is rejected for the present, not as 
being unimportant in itself, but as being irrelevant to 
the immediate question on hand. 



XVII 

GREATEST COMMON MEASURE 

This is the picture of a ceiling of a room. 
Suppose a painter wishes to ornament the edge 
where the ceiling meets the walls, with a little 
simple pattern that will fit exactly into the 
lengths of all the sides, without needing any 
extra corner-ornaments. Suppose he asks us 
to find out for him the length of the longest 




GREATEST COMMON MEASURE 107 

pattern that will fit in exactly. How shall we 
set about it ? 

The pattern is to be repeated over and over 
again. One repeat must end and another begin 
at A ; one must end and another begin at B ; 
the same at C ; and also at D. 




-D 



Let us measure off the length A B from A 
towards C. We will call the end of that 
measure E. Some number of repeats will go 
exactly into A B. We do not yet know what 



108 



LOGIC OF ARITHMETIC 



number of repeats; but some number is to go 
into A B exactly. The same number will go 
exactly into A E. E therefore will be one of 
the points where one pattern ends and another 
begins. Some number of patterns will fit 
exactly into C E. We must find a length that 
will fit some number of times into A E, and some 
other number of times into E C For the present 
we can leave A B alone ; any pattern that will 
fit exactly into A E can be copied exactly into 
A B, because A B is the same length as A E. So 
now we are going to attend to finding a pattern 
that will fit exactly into A E and into E C. 

Let us measure off the length of E C from E 
towards A ; we will call the end of that 
measurement F. So F is another point where 
a pattern begins and ends. Now let us measure 
off the length C E again, from F towards A ; 
and call the end of that measurement G. Then 
G is a point where one pattern ends and another 
begins. We measure off the same length again 
and call the end II ; and again, and call the 
end I ; H and I are ]>oints where one pattern 
ends and another begins. 

If we try to measure the length C E from I 
towards A, we cannot do it ; there is not room, 
1 A is not long enough. 

Any pattern that will fit into I H will fit also 
into H G, and G F and F E, and E C ; because 



GREATEST COMMON MEASURE 109 

all those bits are the same length. So, for the 
present, we will attend only to finding a length 
which can repeat some number of times in I A, 
and some other number of times in I H. Let 
us measure off the length A I from I towards 
H. We will call the end of that measurement 
L. L, then, is one of the points where a pattern 
ends and another begins. Now we will measure 
off the same length, A I from L towards H. 
Ah ! now we find that we have come exactly 
back to H. So that last measurement gives us 
no new points. 

Now let us see what we have got. 

A I, I L, and L H, are all the same length. 
The length of A I has gone twice into I H, 
therefore it will go into H G twice ; and into 
G F, and F E, and E C twice each. So it fits 
exactly into A C, and repeats in A C ? 1 1 times, 
Will it fit exactly into A B ? Yes, it fits exactly 
into A E, and A B is the same length as A E. 

Now we will work this off as a sum on another 
black-board. We will keep this one up, so that 
we can see what we have been doing. 

(If there is no second black-board, the children should 
be told to work the sum off on paper or slates. On no 
account must the diagram be rubbed out till the sum 
has been worked.) 

We ticked off the length A B from A to- 
wards C ; that left E C over. If A B is 18 



110 LOGIC i 


OF ARITHMETIC 


units and A C is 


22 units, how many units is 


EC? Four. Ho* 


' do you know that ? Because • 


18 from 22 = 4. 


We will put that down as 


a subtraction sum. 




22 


= AC 


18 


= AE = AB 


~4 


= EC 


Then we ticked off 4 from the 18. 


18 


= AE 


4 


= EP 


14 


= AP 


4 


= FG 


10 


= AG 


4 


= GH 


~6 


= AH 


4 


= HI 


~2 


= AI 


Then we ticked off 


2 = AI 


= I L from I H 


and this left L H 


without any remainder. 


Then we saw 


that the length A I would 


measure off exactly, without leaving a remainder, 


into all the lengtr 


is H G, G F, F E, E C, because 


they are all equal 


toIH. 


Now we will g( 


> over that in a different way. 



GKEATEST COMMON MEASURE 111 

How often will 18 units go into 22 units ? 
Once, leaving a remainder 4. 

18) 22 (1 
18 

Remainder, used as | 4) 18(4 

new divisor ! 16 

Ditto 2) 4 (2 

4 



Remainder O 

Now, will A I fit exactly into A B ? What 
is A B equal to ? A E. Well, jf the pattern 
fits into A E, will it fit into A B ? And into 
C D ? Yes, because C D is the same length as 
AB. 

And will it fit into B D ? Yes. Because ? 
B D is the same length as AC. 

You remember that the question we started 
with was : — What is the length of the longest 
pattern that will fit exactly into all the lengths 
of the ceiling edge ? We have done that. We 
have done it in three different ways, and we 
know the longest pattern is 2 units long. We 
do not know yet how often the pattern will 
repeat in the room. 

How many times will the length A I fit into 
. the length A C ? 



112 



LOGIC or 


ARITHMETIC 


InAI? 


1 


InlL? 


1 


InLH? 


1 


InHG? 


2 


InGF? 


2 


InFE? 


2 


In EC? 


2 



Eleven times altogether. 
How many times in AB? 

InBD = AC? 
InCD=AB? 

Answer 



11 



9 

11 

9 



40 



The above exercise repeated occasionally preparos 
children for understanding G.C.M. ; • especially if they 
have learned Division as a shortened method for getting 
the result of a series of similar subtractions. It may 
also be given with great advantage to older children 
who have already been badly taught G.C.M. and become 
foggy. In their case it should be followed up by 
a further explanation. Thus : — 

1 Why could you not understand the rule 
before ? I think it was partly for this reason : — 

You have been accustomed to think that the 
most important part of the answer to a Division 
sum is the, quotient ; the remainder matters 
much less. And when you began to do G.C.M. 
you were told to do a Division sum ; and #no 




GREATEST COMMON MEASURE 113 

notice was taken of the quotient ; you were 
made to go on as if the remainder were the 
important thing, and the quotient of no con- 
sequence. And that seemed to you somehow 
dishonest and not quite true ; and it puzzled 
you and prevented your taking in what was 
really going on. But now you can see. When 
the painter comes to paint the ceiling border, 
he must provide paint enough to go all round 
the room. It would not be honest if he only 
painted from A to H. If he sends his assistant 
to block out the points where the patterns must 
begin and end, the assistant must take the 
trouble to go all round and divide up the whole 
edge. If we wanted to find out how many 
points he will have to mark off, we should have 
to take account of the quotients and the re- 
mainders. Missing a quotient would put us 
wrong, even more than missing its remainder. 

But we did not start to paint the room, nor 
to block out the edge, nor to find out how many 
times the pattern would be repeated. We were 
set to find out, first, what length of pattern 
would fit in. For that purpose, you see, the 
remainders matter, and the quotients do not. 

LNow do you see what mistake you made when 
G.C.M. was being explained to you before ? 
Mistakes of just that kind are made about 
many things in life. I am going to tell you 
ir C H 



1H 



LOGIC OF ARITHMETIC 



something that may help you not "to make 
them in future. 

Sit at ease. Shut eyes. Here is a mind- 
picture. 

There is a village. It has water-works and 
a reservoir. At the other end of the village 
there are a few houses which have a deep well 
of their own. A doctor notices that the people 
in those houses do not have a kind of illness 
which is common in the rest of the village. He 
tells the corporation that he suspects the water 
of the reservoir is not wholesome somehow, but 
he does not know exactly what is the matter 
with it. . There is in the village a man who is 
fond of science, and knows a good deal about it. 
The corporation ask this man to try to help 
them to get the water right. He goes to the 
reservoir. What do you see him doing? Does 
he slash in pounds and pounds of stuff at 
random, to try to cure the water before he 
knows what is wrong with it ? No. He has 
a bottle with him. The bottle holds about a 
pint. It is perfectly clean, and has a nice clean 
stopper. He takes a good long look round the 
reservoir. He then dips the bottle in, fills it, 
and stoppers it up. He carries it home. He 
sits down in his study. He puts a drop of the 
water out of his bottle onto a glass slide, puts 
the slide under his microscope, and takes a long, 






long, careful look at it. He turns a little wheel 
round, and looks again and again. 

Then he puts some water out of his bottle 
into a small glass, and adds a little of some 
stuff, and looks, and looks. Then he puts some 
more in another little glass, and adds a bit of 
some other stuff, and looks again. 

In the middle of all this, two neighbours 
come in. One says : ' How can you waste 
your time amusing yourself with your micro- 
scope and all these little glasses that are more 
fit for a doll's house than for a grown man, 
when your neighbours are suffering and dying 
because the water is wrong ? Why don't you 
do something to help us ? ' 

The other neighbour says : ' I heard you 
were employed by the corporation to help the 
town to get the water pure. Why aren't you 
at the reservoir doing it ? Do you think it is 
honest to use only a grain of your stuff and 
purify only a pint- bottle full of water, when you 
are engaged to cure the reservoir? You will 
want a hundredweight of disinfectant, not that 
tiny parcel of it/ 

The scientific man answers : 'When we know 
what is the matter with the water, the corpora- 
tion will send the right people to cure the evil, 
and they must take the right amount of stuff, 
and do the right kind of cleansing. If the} 

H 2 



116 LOGIC OF ARITHMETIC 

attempt to do with less, that will be dishonest. 
But at present we do not know what is the 
matter, nor what treatment the water needs. 
I have been asked to find out these things, and 
am doing my best to find out. When you are 
ignorant, experiment on a small scale at first ; 
that is the way to learn.' 

Have you taken in that picture ? Remember 
it sometimes, when you are tempted to think 
that settling something big, in a hurry, before 
you understand what you are about, will do 
more good than careful, patient study on a 
small scale. 

(All that is on the two black-boards, to be ultimately 
entered in form-books.) 



XVIII 
STANDARD WEIGHTS AND MEASURES 

You remember I told you that in teaching 
a baby to count, it is best to teach him to pack 
up each ten of his counters or pebbles in a box 
or heap to itself. You will find many things 
come easier in your sums, if you take notice 
that a grant deal of what you are learning is to 
pack things up in your minds, so that they 
shall be handy and convenient. It would have 
been inconvenient and troublesome for the 
shepherds of long ago to make a notch for each 
sheep that passed through the gateway into 
the fence ; they packed up the number of sheep 
into tens, and made one stroke — put in the 
tens' place— stand for ten sheep. Later on, 
people packed up tens, and made one stroke in 
the hundreds' place do for ten tens. 

In the same way our pennies are packed up 
into shillings. We need not carry about as 
many pennies as we shall want to spend ; if we 
want to buy twelve penny eggs, we can pay 
one little shilling for the lot ; because the value 
of twelve pennies is packed up into a shilling. 
We do not need to weigh out sixteen 
separate ounces of lard to make the dough for 



118 



LOGIC OF AEITHMETIC 



sixteen small cakes ; sixteen ounce weights are 
packed into one pound weight. 

You will understand sums better if you will 
remember that much of what you hear about 
in the Arithmetic-class is a sort of packing of 
things in your mind or memory. 

Things have to be packed in parcels or 
bundles of different shapes and sizes, according 
to the nature and use of the things themselves. 
Let us think of a few different kinds of mind- 
packing that we know of. 

Ordinary numbers are packed in ? Tens. And 
then into? Tens of tens. That plan was started 
because ? Savages found they had ten fingers to 
count on, before they had names for numbers or 
knew how to write signs for numbers. 

Some things are counted in dozens, such as ? 
Eggs, buttons. Why that is done I really do 
not know ; I have heard that there was once 
a set of people who had a thumb and five 
fingers on each hand ; that gave them ? Twelve 
counters on their two hands. Perhaps these 
men invented counting by dozens, but that 
I cannot say. 

Weights go quite differently — at least in 
England. They are arranged so that each one 
is twice the weight of the one next below, or, 
as it is called, they are arranged in powers of 2. 
For some purposes this is a convenient arrange- 



WEIGHTS AND MEASURES 119 

merit, and it is a very interesting one to learn 
about. 

Things to drink are arranged in something of 
the same kind of way ; in pints and quarts and 
gallons. A pint, I believe, is the size that 
people used to like for their tumblers to drink 
out of. People in Bavaria still use pint tum- 
blers ; it looks very funny to see the children 
lifting great pint pots to their mouths. We in 
England have taken to have our tumblers, only 
half that size ; we call a tumbler, half a pint ; 
and we call as much as two tumblers hold, 
a pint ; and what four will hold, a quart ; and 
so on. 

Can you tell me how inches are arranged? 
In twelves. Twelve inches are called a? 
Foot ; though very few men have feet quite 
twelve inches long. Three twelve-inch feet are 
called a ? Yard. Four inches and a half are 
the length of an average-sized woman's middle 
. finger ; some of you have seen your mothers 
measuring on their fingers. How many ' fingers ' 
make a yard ? Eight. 

Some women measure tape or cloth by hold- 
ing one end of the tape to their noses and then 
stretching their hands back ; that measures off 
a yard. In old books we read of a cubit (that 
is, the length from the tip of the fingers to 
the elbow) ; and a hands-breadth. All these 



120 



LOGIC OF AKITHMETIC 



I 



measurements are rough and not exact, till, 
some Government gives what are called standard 
weights and measures. This saves a good deal 
of disagreement and quarrelling. People in 
shops are made to use weights and measures 
that have been standardized, that is to say, 
stamped by people appointed by the Govern- 
ment ; so that customers can know for certain 
whether they are getting a full pint, or pound, 
or yard. 

Now when a Government is going to stan- 
dardize weights or measures, a good deal of 
thinking has to be done about which are the 
exact sizes that will pack up into each other 
most conveniently. You know what I mean by 
packing, in sums ? I do not mean packing 
actual things in trunks ; I mean packing ten 
ones into a one in the tens' place ; and the 
value of twelve pence in one shilling. Napo- 
leon, more than a hundred years ago, made the 
French people pack all weights and measures 
in tens. The way they measure cloth and 
tapes and roads iB this : — Ten centimetres make 
a decimetre ; ten decimetres make a metre ; 
ten metres make a dekametre, and so on. 
Weights go : — Ten centigrammes make a deci- 
gramme ; ten decigrammes a gramme, and so 
on. That is what is called the decimal arrange- 
ment. We in England have arrangements that 



WEIGHTS AND MEASURES 121 



are not so regular J sixteen ounces make a 
pound ; fourteen pounds make a stone ; twelve 
inches make a foot ; three feet make a yard ; and 
so on. I do not think that either you or I know 
enough about the matter to be able to judge 
which plan is beet on the whole. Everything 
costs something in this world ; whatever is ar- 
ranged, something good and nice and interesting 
will have to be sacrificed to carry it out. The 
thing of most consequence in the matter is that 
we should learn to understand other people, 
and to speak so that they can understand us. 
We ought to know how they weigh and 
measure things, and what they mean when 
they say a pound or a yard ; what we are to 
expect them to give us when we ask for a yard 
or pound. 

If we keep shops by-and-by, we ought to 
know what we are to give each customer. 
Tli is is partly why children go to school ; to 
make them understand other people. People 
used to measure things by ' rule of thumb,' as 
it is called ; every household taking its measures 
from the length of the father's thumb or the 
mother's fingers ; or the tribe took its measures 
from the length of the chiefs foot. Then when 
they sold to each other, there were quarrels ; 
because people's feet or fingers did not measure 
the 6ame length. In civilized countries, people 




122 LOGIC OF ARITHMETIC 

must have standard weights, and all learn to 
use the same ; so whatever the English Govern- 
ment arrange about weights and measures by- 
and-by, I hope you will be ready to learn it 
quickly and use it good-temperedly ; and remem- 
ber that understanding what your neighbours 
mean and do is of far more consequence than 
anything you may happen to think, one way or 
other, about decimal coinage or decimal weights 
and measures. 

And, in order that you may be ready to learn 
new measures easily and use them cheerfully, 
you ought to understand a good deal more than 
most of you do about the different ways in 
which such things pack up, as we called it. So 
one day soon we will have a lesson on packing 
quantities in our sums. 



XIX 

WHAT CAN BE SETTLED BY 
HUMAN LAW 

On Table : — An ounce weight, a pound weight ; 
a yard tape marked off in inches ; a metre ruler marked 
off in centimetres. 

When you see a dozen eggs, or buttons, or 
stamps, you see the whole twelve in the dozen, 
and can count them separately. 




rhen you see a shilling, do you see the 
twelve pennies ? No ; the pennies are not there 
at all, only the cask value of them is in the 
shilling. 

Here is a pound weight. A pound is ? 
Sixteen ounces. Can you see the ounces sepa- 
rate ? No. If I took this to an ironworker's, 
is there anything he could do to make it show 
its sixteen ounce weights ? Yes, he could melt it 
down, divide it into ounces, and stamp each ' one 
ounce ' ; then each would be a proper ounce 
weight, just like this one. If I had sixteen like 
this one, could they be made into a pound 
weight ? Yes ; the ironworker could melt them 
all down together, and make them into a one 
pound weight like this one. 

Is there anything that any one could do to 
a shilling to make it into twelve pennies 1 No. 
Is there any way that any man could make 
a shilling out of twelve pennies ? No. 

So you see there are different sorts of packing 
in our Bums. A dozen buttons is twelve buttons ; 
you can see them and count them separately as 
they lie on their card. 

A pound weight can be made into sixteen 
ounces by melting down and re-stamping ; and 
meantime it already weighs the same as sixteen 
ounces, and contains the same amount of the 
same metal. 



124 



LOGIC OF ARITHMETIC 



A shilling could not be made into twelve 
pennies, and does not weigh them down, and is 
not the same metal as pennies. But the value 
of twelve pennies is packed into a little bit of 
a dearer kind of metal. 

What is packed into this yard measure ? 
Three feet. What is packed into a foot ? 
Twelve inches. So thirty-six inches are packed 
in a yard measure. Here is a French metre 
measure ; what is packed into that ? A hundred 
centi-metres. 

What are these yard and metre measures 
most like : dozens, shillings, or weights ? They 
are like dozens, in so far that we can see the 
inches as they lie along the yard. 

Governments can alter the length of the 
divisions of a yard or metre, in a way that no 
Government could alter the size of an egg. Eggs 
grow, inches are only marked off by men. 

Men might agree to sell eggs in tens instead 
of dozens ; but they could not alter the size that 
the eggs grow by merely marking off differently. 

What are packed into a year ? Months, weeks, 
days. How many days are packed into a year ? 
Three hundred and sixty-five. Who arranged 
that ? Are there the same number of days in 
a year in France as here ? Could the Govern- 
ment alter that? What makes the length of 
a day ? The sun rising and setting, as we call 



it ; which means, you know, that the earth 
turns us to the sun and then away from the 
sun. What makes the year ? Something else 
that happens between the sun and the earth. 
The change from day to night and back to day 
happens three hundred and sixty-five times for 
once that the other changes happen, the ones 
that make a year. 

Who arranged that ? Was it the king ? Was 
it the Parliament ? Was it Napoleon or the 
French Government ? Do you think that if all 
these great people agree together and decided 
to alter the number of days in a year, they could 
do it? No. Long ago a number of learned 
men spent time on finding out how many days 
there are in a year. It took them a very long 
time to do, but they found out at last. But as 
for altering the number, oh, no ! no one has 
ever been able to do that. 

You see how many different things had to be 
thought about in arranging the tables in your 
sum-books. The number of fingers we have; 
the most convenient way to divide lengths ; the 
most comfortable-sized tumbler to hold ; the 
most convenient coins to carry about ; the length 
of days and years ; all sorts of things have been 
taken into account at dhTereut times. So the 
least we can do is to take things as they come, 
make the best of them, and learn to understand 



4 



126 



LOGIC OF ARITHMETIC 



Nature, and each other's needs and wishes ; and 
to fit in amongst it all as best we can. 

Now sit at ease, shut eyes, and make mind- 
pictures. 

A king sitting in a Parliament House. He 
says : ' The length of my thumb-joint shall be 
called an inch; and thirty -six inches shall be 
called a yard.' London shopkeepers now go to 
an office at the Guildhall, and take tapes to be 
marked in inches. 

A French general in a cocked hat, with a 
uniform and sword. He says : ' A metre shall 
be divided into a hundred centimetres.' French 
shopkeepers go to an office in Paris, and get their 
metre tapes measured. 

Another picture. Now the king and the 
general sit waiting. Learned men come and 
say : ' There are three hundred and sixty five 
days in a year.' The king and the general bow 
their heads. The almanack-makers write down 
what the learned men said \ 

1 There are 365 days and a few hours in a year. But 
it is not advisable to enter on details in the present 
lesson. 




On Table : — A £5 note and a £10 note. 

I hope you have begun to see that Arithmetic 
is very much concerned with the question of 
packing things in convenient shapes and sizes, 
to carry them in our understanding and our 
memories : sometimes even with convenient 
packing of things to carry in our hands or 
pockets. Buttons are sewn in dozens on cards 
to avoid the need of counting them at busy 
times, just as grocers put up sugar in pounds 
ready for customers. Eggs are ready packed 
in boxes so that a whole dozen can be handled 
as one boxful. We pack up enough iron or 
braes to make sixteen ounce weights into one 
lump, which we call ? A pound weight. And 
so on. If you think, you can see that, if an 
ancient shepherd made a notch for every ten 
sheep that had been counted, he was really 
doing something of the same kind. In the 
same way we put a mark for a foot at the end 
of every twelve inches, or a mark for a yard at 
the end of thirty-six inches. 

When we take a shilling in our purse instead 
of twelve pennies, the packing is of a different 





kind We do not pack the twelve pennies into 
one lump ; we do not take the pennies themselves 
in any shape. Do you remember what it is that 
is packed into the little shilling which we use 
instead of the twelve pennies ? The value of 
twelve big clumsy copper pennies is packed into 
one little bit of the more valuable metal, silver. 

In multiplication, when we say 6 times 8 are 
48, we pack into a short form the business of 
adding six eights, one after the other. When 
we divide 37 by 5, what we are really doing is 
to see how often we can subtract 5 from 37 ; 
instead of doing the subtractions one after 
another, we pack the whole lot together and do 
it all in one sentence. If you think about it 
sometimes, as you go on learning, you will find 
that a great deal of your Arithmetic is simply 
a sort of packing up to make short cuts. We 
get the value, so to speak, of a long roundabout 
proceeding, packed into a convenient bit of 
work. 

Well now, let us go back to our twelve 
pence and our shilling. You perfectly under- 
stand that the reason why this shilling is worth 
as much as two sixpences is because there is 
twice as much silver in it as in one sixpence. 
It weighs twice as much. And the reason why 
the silver is worth as much as twelve pennies is 
because it is a more costly material. Can you 




tell me any other instance of one coin being 
worth a great many others because it is of 
a more costly material ? A sovereign is worth 
twenty shillings or eight big half-crowns, be- 
cause gold is much more costly than silver. 
How many pennies is a sovereign worth ? 240. 
Do you think you quite clearly understand why? 

Well now, look at these two pieces of paper. 
What are they worth ? One is worth £5, the 
other £10. Why ? Is paper a more costly 
material than gold ? No. You know that it is 
very much cheaper by weight. And it would 
take a great many of such papers as these to 
weigh down a sovereign. I am sure that these 
papers are not worth in themselves a penny 
apiece as paper. And they are exactly alike ; 
that is, the paper part of them is alike ; one is 
not bigger or thicker than the other. Yet 
I could get five sovereigns for one and ten for 
the other. Why is that ? What makes them 
so valuable ? And why is one worth twice as 
much as the other 1 What does it all mean ? 
Something must be packed into a little piece of 
paper, to make it worth so many sovereigns. 
Let us see if we can find out what is 
packed into these bits of paper to make them 
valuable. 

It is never worth while guessing at the 
meaning of a thing until after you have looked 

I 



130 



LOGIC OF ARITHMETIC 



at all that there is to be seen in it, and found 
out all that there is to learn about it. So we 
will begin by reading all that is on these papers. 
And then we shall see if we can make out what 
it is that has been packed into them. 
Bank of England. 

I promise to pay the Bearer on Demand the 
sum of Five Pounds. 

London 5 Sept. 1 902 

For the Gov 1 " 8 and Comp* of the 
Bank of England. 
J. G. Naime, Chief Cashier. 

The Bank of England promises to pay to 
whoever brings this piece of paper to the Bank, 
£5. And people feel so sure that the Bank will 
pay the five pounds that anybody will give five 
pounds for the paper. They feel they will always 
get their money back. 

How did people come to feel so sure that the 
Bank would keep its promises ? Because it has 
gone on so many generations always keeping 
them. 

In some countries you may see written on 
a Bank note a promise to pay five dollars, 
and you may hear people bargaining as to 
whether that promise (or that bit of paper) is 
worth two dollars or three. Which means that 
taking paper money in these countries is more 
or less a sort of gambling or speculation. People 



PAPER MONEY 131 

do not feel sure enough that the promise will 
be kept to give the full value for the piece of 
paper that it is written on. 

What, then, is it that is packed into these 
pieces of paper that we call Bank notes ? A 
habit of trust in the honour of the Bank of 
England. 

We have been hearing a good deal of late 
years about the honour of England and the 
glory of England, and there have been pro- 
cessions and shows and military displays, which 
are supposed to be for the honour of the country. 
I should like you to notice, my dears, that any 
country can wave flags and bang drums and ring 
bells and walk about in processions and shout 
about its honour and glory, if it pleases. The 
real glory and honour of England are much better 
expressed, it seems to me, by the fact that, if 
its National Bank promises to pay £5 or £500, 
the people of the country take that promise for 
its full value. They believe that their Govern- 
ment will keep its promises. 



J 2 



XXI 
THE DOG'S PATH 



If children are to learn Curve-Tracing, or any sort 
of Analytical Geometry, the following exorcise forms 
a good introduction to the subject, as it puts their 
minds into line with the alternation of tendencies, the 
apparent conflict of forces, by which physical Nature 
and human evolution are alike worked out. It obviates 
any sense of mystery about what the whole thing means. 
Once a child has grown accustomed to pay attention 
alternately to the aspirations of the dog and to that of 
the rabbit, and to see a curve growing up under the 
impulsion of his own alternating sympathy, it is then 
legitimate to tell him that there are other forces, besides 
the wishes of animals, at work making curves, and that 
he can learn to follow the action of many of these, even 
when he can trace no conscious motive which is setting 
them to work. 

Try to fancy that this black-board is a field. 
It has a brick wall all round it, with no opening 
except at A, where there is a gate. 

At Z there is a rabbit hole. A rabbit came 
out into the field at Z and wandered about till 
he came to here (write the figure l), where he found 
something he liked to eat. After a little while, 
a dog came in at the gate A ; the rabbit caught 
sight of him, and directly afterwards the dog 
caught sight of the rabbit. Now let us try if 
we can make out what happened. First we 
must try to think what the animals would each 



THE DOG'S PATH 133 




134 



LOGIC OF ARITHMETIC 



like to have happen ; the rabbit saw the dog first ; 
what do you think he wished ? To get away. 
Perhaps his first idea is to run straight away 
from the dog. But he can't ; the wall prevents 
him. What will he do next ? If he was a very 
foolish rabbit, he might stand trying to get 
through the wall, and fussing, and saying : — ' Oh ! 
dear ! I must get out here, there is no other 
way. Oh ! this cruel wicked wall ! it is pre- 
venting me from getting out ! Oh 1 what shall 
I do ? This is the straight way out ; and I 
must knock a hole in this wall ! Oh ! what 
shall I do ? ' 

If the rabbit was a foolish rabbit, he might 
go on in that way, till the dog caught him and 
ate him up. 

But we will suppose he is a sensible rabbit 
who is not fond of knocking his head against 
brick walls, and who has learned to use his 
brains properly about his own business. What 
do you think he will think of next ? He will 
think of trying to get to his hole. He would 
like to jump straight to his hole ; but he cannot 
go right across a field in one jump. However, 
we will draw a line to show what he would like 

to do. (Draw ihe lino 1 Z.) 

But just when the rabbit gave up the notion 
of getting out through the wall at 1, the dog 
saw him. The dog had his little wishes about 



THE DOG'S PATH 



135 



the rabbit. What do you think he wished ? 
What would the dog like to do ? Jump on the 
rabbit and kill him. The dog can no more get 
across a big field in one jump than the rabbit 
could ; but he would like to do so. We will 
draw a line to show what the dog thinks he 

would like to do. (Draw the line A I.) As the 

dog cannot jump from A to 1 he jumps as far 
as he can ; his first jump takes him to B. But 
by the time he has got to B, he sees the rabbit 
has got to 2. Do you think he will go on 
scampering down the line A 1 now he sees the 
rabbit is not at 1 , but at 2 ? Of course not. He 
will wish now to jump from B to 2. But that 
again- is too far for him to go in one jump ; he 
jumps to C, but by the time he gets there the 
rabbit is at 3. 

The teacher should go on step by stop, drawing the 
successive wishes of the dog, and marking off the jumps 
of both animals, taking care to keep up, all the time, the 
children's consciousness of what the animals are each 
thinking and doing; till the diagram has evolved itself 
on the black-board. Then : 

Now tell me, what do all these lines repre- 
sent? The line 1 to Z represents the path 
which the rabbit would like to go along in one 
jump, and does take in eighteen jumps. The 
lines A 1, B 2, C3, and all the other straight 
lines, each represent a line that the dog at 
some moment wished to jump along ; he jumped 



md then changed his mind 
and jumped a bit of the next, and so on. We 
drew all those straight lines ; you saw me draw 
them by the ruler, did you not ? 

But here is a curved line A to Z. Who drew 
that ? I drew no line except straight ones by 
the ruler. Look at it well. Make sure that 
you see it and all the lines on the black-board. 

Now sit slack, shut eyes, and think what the 
curved line is and how it came. 

Open eyes and sit up. What is the curved 
line ? The path which the dog really ran, when 
at each step he meant only to go down some 
straight line. 



XXII 
THE BALL'S PATH 



You remember that we drew the path of a 
dog in a field, by drawing straight lines to show 
what he wished to do but could not. The dog 
was dragged several ways, first one way and 
then another. The dog was dragged only by 
his own wishes and thoughts. 

But a thing which has no wishes or thoughts 
of its own, may be dragged by other forces. We 
call what makes things move, force. Well, as 



THE BALL'S PATH 137 

I said, things may be moved by other forces 
besides their own thoughts and wishes. If you 
throw a ball, you make it go along. We do not 
know that it has any wishes of its own ; your 
wish is a force which moves it. But your wish 
is not the only force which moves it ; for if you 
aim exactly at the top corner of the house, the 
ball will not hit exactly the top corner; some 
other force, what we call its weight, pulls it a 
little downwards as it goes along ; so, between 
the two forces, the ball makes a curve ! When 
a thing moves in a curve, it is usually because 
two forces, or more than two, are pulling or 
pushing it in different directions. 

You are going to begin learning about curves. 
It will help you to keep out of many muddles, 
if you will try to remember that, when you see 
a curve in a book or on paper, it represents 
some real form or movement ; or something more 
or less like a real form or movement. It may 
be simpler than the real path or shape ; but it 
is only simpler ; it is really more or less like 
something meant to be real. 

But when you see straight lines, they are 
seldom meant for anything real. A straight 
line represents either a path that one force alone 
would have taken something along if no other 
force had interfered ; or else it is just put in 
for convenience, to measure by. You have per- 






138 LOGIC OF AKITHMETIC 

haps seen tailors' fashion books, with directions 
for taking measures. You see a picture of a 
man with a coat on, and straight lines drawn 
across the shoulders or bust. You would be 
dreadfully puzzled if you thought of those lines 
as parts of the picture ; because no coat has 
lines across the shoulders or bust ; but you 
know they are meant, not for seams in the coat, 
but for a measuring-tape supposed to be stretched 
across the man in order to measure the width 
of his coat. I have known children puzzled out 
of their wits, and never able to understand their 
Geometry for years of their school-time, because 
they mistook straight lines for real parts of 
some curved thing. 

And if you ever feel worried and puzzled over 
your Geometry lessons, shut your eyes for a 
minute and think about the rabbit and the dog. 
Then open eyes, and look at your book ; and say 
to yourself, 'Which parts of this picture are 
meant to be real like the real path of the dog ; 
and which parts are only like the straight lines 
that teacher drew on the black-board on purpose 
to make us children understand how the dog 
came to run in a path in which at first he did 
not mean to go?' 



XXIII 

EXERCISE TO PREPARE FOR 
GENERAL FORMULAE 

1 and 3 are? 4. And 5? 9. And 7? 16. 
And 9 ? 25. 

And so on, adding successively the odd numbers. 
The results should be entered in a column on the right 
hand of the black-board, and ultimately on the right 
hand of the page in the formula-books. On the left 
hand should be entered successively : — 

1x1 = 1 

2x2=4, &c, thus :— 

1x1=1 1 

3 
2x2=4 4 

5 
3x3=9 9 

7 
4x4 = 16 16 

9 

5x5 = 25 25 

&c. &c. 

This exercise should foe repeated occasionally, as 
mere practice in adding and multiplying, till the fact 
of the identity of results of the two processes has been 



140 



LOGIC OF AKITHMETIC 



worked into the children's consciousness, simply as 
a fact, empirically observed, and, as yet, unexplained. 

Then, they Bhould bo told to build up the successive 
odd numbers, either with cubes, or on squared paper (a). 

They should be led to see that, by building 3 round 
1, they get a square block of the same size and shape as 
by building 2 on 2 ; 5 round the block of 4 gives a block 
the same size and shape as three rows of three, and 
so on. 

These things should be done simply as exercises in 
adding, multiplying, and neat pencil-shading or chalking. 
They should each be repeated many times. Great 









':/ 








[/Pi 






— -i 


n 










2 


3 




















1 1 








6 




































-- 













4 

5 

_ 



accuracy should be insisted on ; but no explanation 
should be attempted. The good mathematician will see 
that none is necessary. An ordinary teacher who 
attempted one would be almost sure to eonfuse and 
mislead. The important thing to secure is that the 
children's eyes, fingers, and attention should be occupied 
about the diagrams, thus lodging clear images of them 
on the memory, ready to crop up on some future 
occasion ; as explained in my Address On the Prepara- 
tion of the Unconscious Mind for Science '. 

In case the pupil afterwards takes up Algebra or the 
Calculus, the theory of General (or Sliding) Formulae 

1 Parent's Review, 1809. 



FOKMULAE 141 

will, when it has to be explained to him, find ready 
a substratum of observed and familiar fact, illustrating 
one such Formula : — 

(» + l) a = n s +2n + l 
The lack of any such basis, the consequent difficulty of 
making the pupil grasp the idea of a General Formula, 
forms one of the great obstructions to sound progress in 
Mathematics. 

But, even if he is never to go beyond ordinary 
Arithmetic, no harm will have been done. He will 
have gained a little practice, useful in itself, in the 
comparison of numerical results arrived at by different 
methods. The time spent on gaining familiarity with 
the Genesis of Tables of Squares will not have been 
wasted. Arithmetic itself is always best taught, when 
it is taught on methods which constitute a sound prepara- 
tion for the study of the higher Algebra. 



APPENDIX 

I have not succeeded in drawing out any 
quite satisfactory scheme for suggesting the 
feeling of proportion to the imagination of 
children who have never used any optical 
instrument. Perhaps some reader may devise 
one. For children accustomed to use a magni- 
fying-glass, the following mode of conveying 
the idea is simple. Draw on paper two or three 
short but unequal lines, and mark each with 
a letter. Let the children look at each, first 
with the naked eye, then with a magnifier. 
Tell them to draw (or to suppose drawn) the 
lines as seen through the glass. Then say that 
the little line marked (a) is to the big line 
marked (a) in the same proportion as the little 
line marked (b) is to the big line marked (6), 
and as the little line marked (c) is to the big 
line marked (c). Draw also a small triangle 
and quadrilateral, and a copy of. each as seen 
through the magnifier ; say that the little 
triangle is to the big triangle in the same 
proportion as the little quadrilateral to the big 
one. Then show (a) through a weak magnifier 



APPENDIX 



143 



and (b) through a stronger one ; and say that, 
now, the pairs of lines are not in proportion to 
each other. 

I would again emphasize the caution that 
the lessons in this book are mere specimen 
types, intended, not to be slavishly read aloud 
to a class, but to suggest methods, such as 
I have found useful, of dealing with various 
arithmetical difficulties. On p. Ill a sum is 
put in the form kindly shown me by an 
elementary-school teacher, as the one familiar 
to the pupils to which she is accustomed. 
There is a different form more suitable for 
advanced pupils doing long sums. Each teacher 
should use the form to which his pupils are 
accustomed. 

On p. 61, the lesson is given as suited to 
children accustomed to begin multiplying by 
the unit figure of the multiplier ; any teacher 
who habitually begins at the other end of the 
multiplier should modify the lesson to suit his 
class. 

When I am giving the lesson on G. C. M., 
I usually connect it with a story of a painter in 
Home, who knew nothing about our system of 
notation, and who, having to decorate a ceiling, 
asked his children to spare his labour by mark- 
ing off, on a plan, as many as they could of the 
points where he would have to make a pattern